Design and workspace analysis of a 3-degree-of-freedom parallel swivel head with large tilting capacity

Abstract

Traditional parallel mechanisms are usually characterized by small tilting capability. To overcome this problem, a 3-degree-of-freedom parallel swivel head with large tilting capacity is proposed in this article. The proposed parallel swivel head, which is structurally developed from a conventional 3-PRS parallel mechanism, can achieve a large tilting capability by means of structural improvements. First, a modified spherical joint with a maximum tilting angle of ±120° is devised to diminish the physical restrictions on the orientation workspace. Second, a UPS typed leg is introduced for the sake of singularity elimination. The superiority of the proposed parallel swivel head is theoretically proved by investigations of singularity-free orientation workspace and then is experimentally validated using a prototype fabricated. The theoretical and experimental results illustrate that the proposed parallel swivel head has a large tilting capacity and thus can be used as swivel head for a hybrid machine tool which is designed to be capable of realizing both horizontal and vertical machining.

Introduction

Recently, parallel machine tools are widely used in manufacturing industries due to their potential advantages over the serial ones. These advantages, such as low inertia, high stiffness, and high precision, make parallel machine tools quite suitable for high-speed machining of thin-walled workpieces and freeform surface parts.^1–3 In the early days, parallel machine tools are mainly developed in fully parallel form, ⁴ which is characterized by a relative small workspace with plenty of singularities and lower tilting capacity. These defects strongly restrict the applications of the inchoate parallel machine tools. One promising solution to this problem is the hybrid machine tool which is composed of both serial and parallel mechanisms. This idea has been proved practically by many commercial products, such as the Tricept,^5,6 Ecospeed, ⁴ and Exechon. ⁷ Hybrid machine tools properly take full advantage of both the serial mechanism and parallel mechanism: the serial part is beneficial to workspace enlargement while the parallel part is helpful to stiffness and accuracy improvements.

Among the research activities in workspace of the hybrid machine tool, studies mostly focus on the reachable workspace of the mounted parallel swivel head. In the literatures, the workspace of the parallel swivel head (PSH), such as Sprint Z3 head, has been extensively studied under prescribed constraints including singularities, motion range of actuators, rotation constraints of revolute and spherical joints.^8,9 The behavior of singularity for parallel mechanisms is more complicated than that of serial ones. In general, singularity usually decreases the volume of feasible workspace^10–12 because parallel robot will become unstable, or even uncontrollable near those singular areas. The workspace degradation caused by singularities also has negative impacts on singularity-free path planning^13,14 and workspace determination. ¹⁵ However, it is believed that actuation redundancy is one of the feasible ways to alleviate the problems caused by singularities. ¹⁶

The Ecospeed by DS-Technology, which is actually designed on basis of a 3-PRS PM, originally has no actuation redundancy and is easily influenced by singularities. Thereby, the two spindle tilt angles can only achieve a maximum value of ±40°, which is unable to meet the posture requirements in practical machining of those parts with complex freeform surface, such as impeller and blisk. In this article, a 3-degree-of-freedom (DOF) PSH is developed on the basis of a 3-PRS PM. Modified spherical joints with large cone angle are adopted instead of the common spherical joint or universal joint. Actuation redundancy is introduced by mounting an additional UPS kinematic chain between the moving platform and fixed base in order to achieve a large tilting capacity.

The remainder of this article is organized as follows: the design conception is expatiated in section “Conceptual design of the PSH.” In section “Kinematics analysis,” kinematics analysis is performed. Jacobian matrix is first generated in section “Jacobian matrix generation” and then in section “Singularity-free orientation workspace,” singularity-free orientation workspace is obtained and discussed. Experimental results are presented in section “Prototype and experiments.” Finally, conclusions are given in section “Conclusion and future works.”

Conceptual design of the PSH

Design objective

According to the research background mentioned above, the PSH to be designed is required to achieve a maximum tilting angle of ±90° around its both two rotatory axes in either direction. It is significant, especially, to those machine tools designed for machining of aircraft components, such as impeller, blisk. From the point of control and orientation workspace, singularity avoidance and physical constraints should be taken into account respectively. Consequently, the proposed PSH can be derived by making several significant improvements to a classical 3-PRS mechanism.

Architectural description

As shown in Figure 1, the proposed PSH is developed by adding a UPS leg to a 3-PRS parallel mechanism, because the UPS leg will not introduce any constraint wrench to the moving platform. The proposed PSH consists of a moving platform, a fixed base, three identical PRS kinematic legs and one UPS kinematic leg. Each PRS leg connects the moving platform to the fixed base with a modified spherical (S) joint followed by a revolute (R) joint and a prismatic (P) joint in sequence, where P joint is the active joint. The P joint is achieved by a servo-motor-driven lead screw system. To enlarge the singularity-free workspace of the PSH, a redundant leg is introduced and mounted between the central points of the moving platform and the fixed base. The redundant leg connects the moving platform to the fixed base with a modified S joint, a P joint, and a universal (U) joint in sequence, where P joint is also the active joint.

Figure 1.

Architecture overview of the proposed PSH.

As a lower mobility PM, a 3-PRS manipulator can provide 3 DOF, including a translational motion and two rotary motions. ⁸ The calculation result given by the general Grübler–Kutzbach criterion, as shown in equation (1), indicates that the proposed PSH has the same mobility with a 3-PRS manipulator

$$Frd=\lambda \left(n-g-1\right)+{\displaystyle \sum _{i=1}^{g}fr{d}_i}=6\times \left(10-12-1\right)+21=3$$

(1)

To realize 5-axis machining, the proposed PSH can be mounted on a serial X-Y translational platform, as shown in Figure 2, where the serial platform is responsible for translational motion of the 5-axis machine tool, especially for compensating the parasitic motion caused by the PSH. ¹⁷

Figure 2.

Realization of 5-axis machine tool with the proposed PSH.

In workspace design of a practical manipulator, cone angle limits of spherical joints always greatly influence the volume of the workspace. In general, it is impossible for a common spherical joint to realize a cone angle larger than 90°; thereby, the Hook-type spherical joint is always adopted instead. ¹⁸ However, in practice, due to its shortage in stiffness, the Hook-type spherical joint is always fabricated in a large size to meet the stiffness requirements, ¹⁹ which will introduce other undesired degradations, such as interferences. In this article, a modified spherical joint, as shown in Figure 3, is proposed and used instead of the common one.

Figure 3.

Modified spherical joint and its orientation workspace: (a) the solid model and (b) the prototype.

The modified spherical joint consists of a bearing stud, two upper ball-sockets, a lower ball-socket, a bearing and a bearing chock. As depicted in Figure 3, a deep groove is cut through the upper and low ball-sockets to endow the ball-end bearing stud with a large tilting capacity. The lower ball-socket is mounted on a bearing chock via a taper roller bearing thereby the spin angle of the modified spherical joint has no limits and still three available DOF. But, the stiffness of this modified spherical joint is still relatively lower than it of common one and there will be an additional bending moment on its bearing, so a precise verification is required in its design. Because of the compact structure, relatively high stiffness comparing with Hook-type spherical joint and, especially, the large tilting angle of 120°, the modified spherical joint is still inclined to be utilized, which significantly enlarges of the orientation workspace of the S joint.

Kinematics analysis

The vectors and reference frames are depicted in Figure 4. Reference points ${\mathit{A}}_i$ (i = 1, 2, 3) and ${\mathit{B}}_i$ (i = 1, 2, 3, 4) denote the installation position of the ith vertical linear actuator on the fixed base plane and the central point of the ith modified spherical joint, respectively. Points ${\mathit{C}}_i^{\prime }$ and ${\mathit{C}}_i$ (i = 1, 2, 3) indicate the position of the slider along the sliding rail and the position of the R joint, respectively.

Figure 4.

Vectors and position representation of the PSH.

For the sake of kinematics analysis, a fixed Cartesian reference coordinate frame $\mathit{O}\{{\mathit{x}}_0,{\mathit{y}}_0,{\mathit{z}}_0\}$ is set on the fixed base at point O , the central point of triangle $\mathrm{\Delta }{\mathit{A}}_1{\mathit{A}}_2{\mathit{A}}_3$ , and x₀-axis passes through ${\mathit{A}}_1$ and z₀-axis is along its normal vector. Similarly, a moving reference frame $\mathit{P}\{\mathit{u},\mathit{v},\mathit{w}\}$ is attached on the moving platform at point P ( B ₄), which is the central point of triangle $\mathrm{\Delta }{\mathit{B}}_1{\mathit{B}}_2{\mathit{B}}_3$ , and u-axis passes through ${\mathit{B}}_1$ as well as w-axis is along its normal vector.

In practice, three PRS legs are symmetrically mounted about the central axis of the proposed PSH. The redundant UPS leg is mounted between O and B ₄ , and let ${\mathit{d}}_4=\overrightarrow{{\mathit{OB}}_4}$ . Meanwhile, $r_a=|\overrightarrow{{\mathit{OA}}_i}|$ denotes the distance from O to ${\mathit{A}}_i$ , $r_b=|\overrightarrow{{\mathit{PB}}_i}|$ indicates the distance between P and B_i in the ith PRS leg, and $r_m=|\overrightarrow{{\mathit{PB}}_4}|$ is distance from point P to B₄ in the UPS leg. Relating to fixed frame $\mathit{O}\{{\mathit{x}}_0,{\mathit{y}}_0,{\mathit{z}}_0\}$ , some vectors are also defined, including $\mathit{p}=\overrightarrow{\mathit{OP}}={\left[\begin{array}{c}\hfill p_x{p}_y{p}_z\hfill \end{array}\right]}^T$ , ${\mathit{h}}_i=\overrightarrow{C{\prime }_i{C}_i}$ , ${\mathit{l}}_i=\overrightarrow{{\mathit{C}}_i{\mathit{B}}_i}$ , ${{\mathit{d}}_{\mathit{i}}|}_{i=1,2,3}=\overrightarrow{{\mathbf{A}}_i{\mathbf{C}\prime }_i}$ . ${\mathit{a}}_i=\overrightarrow{{\mathbf{OA}}_i}$ and ${\mathit{b}}_i=\overrightarrow{{\mathbf{PB}}_i}$ . ${\mathit{b}\prime }_i$ denotes ${\mathit{b}}_i$ expressing in $\mathbf{P}\{\mathit{u},\mathit{v},\mathit{w}\}$ .

Constraint conditions and parasitic motions

The proposed PSH, which is actually developed on the basis of a 3-PRS PM, always has three parasitic motions: one rotation about z₀-axis of the fixed frame and two translations along x₀- and y₀-axes of the fixed frame. ¹⁶ In fact, those parasitic motions are generated synchronously with the other independent motions under the mechanical constraints imposed by the R joints.

In Figure 4, the position vectors of ${\mathit{A}}_i$ (i = 1, 2, 3) and ${\mathit{B}}_i$ (i = 1, 2, 3) with respect to frames O and P can be expressed as

$${\mathit{a}}_1={[r_a,0,0]}^T,{\mathit{a}}_2={[-r_a/2,\sqrt{3}{r}_a/2,0]}^T,{\mathit{a}}_3={[-r_a/2,-\sqrt{3}{r}_a/2,0]}^T$$

(2)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{b}\prime }_1=[r_b,0,0{]}^T,{\mathit{b}\prime }_2=[-r_b/2,\sqrt{3}{r}_b/2,0{]}^T}\\ \multicolumn{1}{c}{{\mathit{b}\prime }_3=[-r_b/2,-\sqrt{3}{r}_b/2,0{]}^T,{\mathit{b}\prime }_4=[0,0,-r_m{]}^T}\end{array}$$

(3)

In this work, a ZYX Euler angle about the fixed frame is used to describe the orientation of the moving platform and the rotation matrix can be described as

$$\mathit{T}={\mathit{T}}_{zyx}={\mathit{R}}_{z\alpha }{\mathit{R}}_{y\beta }{\mathit{R}}_{x\gamma }=\left[\begin{array}{ccc}c\alpha c\beta & c\alpha s\beta s\gamma -s\alpha c\gamma & c\alpha s\beta c\gamma +s\alpha s\gamma \\ s\alpha c\beta & s\alpha s\beta s\gamma +c\alpha c\gamma & s\alpha s\beta c\gamma -c\alpha c\gamma \\ -s\beta & c\beta s\gamma & c\beta c\gamma \end{array}\right]$$

(4)

where c_* and s_* correspond to cos(_*) and sin(_*), respectively. As shown in Figure 4, the position vector ${\mathit{q}}_i=\overrightarrow{{\mathit{OB}}_i}=[\begin{array}{c}\hfill q_{\mathrm{ix}}{q}_{\mathrm{iy}}{q}_{\mathrm{iz}}\hfill \end{array}{]}^T$ can be given as

$${\mathit{q}}_i=\mathit{p}+{\mathit{Tb}\prime }_i$$

(5)

Considering the physical constraints imposed by the R joint, the B_i (i = 1, 2, 3) can only move in the leg plane. The following three constraint equations can be yielded

$$q_{1y}=0$$

(6)

$$q_{2y}=-\sqrt{3}{q}_{2x}$$

(7)

$$q_{3y}=\sqrt{3}{q}_{3x}$$

(8)

Substituting equations (4) and (5) into equations (6)–(8) yields

$$\alpha ={\tan }^{-1}[s\beta s\gamma /(c\beta +c\gamma )]$$

(9)

$$p_x=\frac{r_b}{2}(c\alpha c\beta -s\alpha s\beta s\gamma -c\alpha c\gamma )$$

(10)

$$p_y=-r_{b}s\alpha c\beta$$

(11)

Hence, according to equations (9)–(11), the three parasitic motions of the proposed PSH can be derived from the two given independent rotations. The parasitic motion along z _o-axis is identical with the spindle rotation axis and the other two can be compensated by auxiliary serial X-Y translational platform; therefore, the PSH is feasible to be utilised as the swivel head of 5-axis machine tool.

Inverse kinematics

With reference to Figure 4, a vector-loop equation can be written for the ith PRS leg

$${\mathit{l}}_i+{\mathit{a}}_i+{\mathit{d}}_i+{\mathit{h}}_i={\mathit{p}+\mathrm{Tb}}_i$$

(12)

Considering three sliders are only allowed to move downward, solving equation (12) leads to the inverse kinematics for the ith PRS leg

$$\begin{array}{c}\multicolumn{1}{c}{\left|{\mathit{d}}_i\right|=m_{\mathrm{iz}}-\sqrt{{\left|{\mathit{l}}_i\right|}^2-m_{\mathrm{ix}}^2-m_{\mathrm{iy}}^2}}\\ \multicolumn{1}{c}{{\mathit{m}}_i=[m_{\mathrm{ix}},m_{\mathrm{iy}},m_{\mathrm{iz}}{]}^T=\mathit{p}+{\mathit{Tb}}_i-{\mathit{a}}_i-{\mathit{h}}_i}\end{array}$$

(13)

For the redundant leg, the inverse kinematics can be simply expressed as equation (14), and four actuations of the PSH are immediately determined

$$\left|{\mathit{d}}_4\right|=|\mathit{p}+{\mathit{Tb}}_4|$$

(14)

Jacobian matrix generation

Referring to Figure 4, velocities of the four S joints can be represented by velocity of the moving platform as equation (16) and the motions of the ith leg as equation (17)

$${\mathit{V}}_{\mathrm{Si}}={\mathit{v}}_p+{\mathbf{\omega }}_p\times {\mathit{b}}_i$$

(15)

$${\mathit{V}}_{\mathrm{Si}}={\stackrel{·}{\mathit{d}}}_i+{\mathbf{\omega }}_{\mathrm{li}}\times {\mathit{l}}_i$$

(16)

where “×” indicates the cross product between vectors, ${\mathit{V}}_{\mathrm{Si}}$ is velocity vector of the ith S joint, ${\mathit{v}}_p=[v_x,v_y,v_z{]}^T$ , and ${\mathbf{\omega }}_p=[{\omega }_x,{\omega }_y,{\omega }_z{]}^T$ denote the linear and angular velocity of the moving platform, respectively. ${\stackrel{·}{\mathit{d}}}_i$ denotes the velocity of the ith linear actuator, and ${\mathbf{\omega }}_{\mathrm{li}}$ indicates the angular velocity of the ith leg.

Let ${\mathit{k}}_i=[k_{\mathrm{ix}},k_{\mathrm{iy}},k_{\mathrm{iz}}{]}^T$ be the unit vector of $\overrightarrow{{\mathit{C}}_i{\mathit{B}}_i}$ , ${\mathit{u}}_i=[u_{\mathrm{ix}},u_{\mathrm{iy}},u_{\mathrm{iz}}{]}^T$ be the unit vector of rotational axis of the ith R joint, $\stackrel{·}{\mathit{d}}=[\begin{array}{c}\hfill {\stackrel{·}{d}}_1{\stackrel{·}{d}}_2{\stackrel{·}{d}}_3{\stackrel{·}{d}}_4\hfill \end{array}{]}^T$ indicate the velocity vector of the four actuations and $\stackrel{·}{\mathit{q}}=[\begin{array}{c}\hfill v_z{\omega }_x{\omega }_y\hfill \end{array}{]}^T$ denote the velocity of moving platform. Note that ${\mathit{k}}_i·{\mathbf{\omega }}_{\mathrm{li}}=0$ and ${\mathit{u}}_i·{\mathit{V}}_{\mathrm{si}}=0$ . Taking equations (15) and (16) and dot-multiplying both sides with k _i , the Jacobian matrix of the PSH, which presents the relation between the velocity of moving platform and that of four actuations, in singularity-free workspace can be derived

$$\stackrel{·}{\mathit{d}}={\mathit{J}}_a{\left[\begin{array}{ccc}\hfill v_z\hfill & {\omega }_x& {\omega }_y\end{array}\right]}^T$$

(17)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{J}}_a={\mathit{B}}^{-1}\mathit{A}}\\ \multicolumn{1}{c}{\mathit{A}={\mathit{K}}_1+{\mathit{K}}_2{\mathit{K}}_4^{-1}{\mathit{K}}_3}\\ \multicolumn{1}{c}{\mathit{B}={\left[\begin{array}{cccc}\hfill {\mathit{d}}_1·{\mathit{k}}_1\hfill & 0& 0& 0\\ \multicolumn{1}{c}{0}& {\mathit{d}}_2·{\mathit{k}}_2& 0& 0\\ \multicolumn{1}{c}{0}& 0& {\mathit{d}}_3·{\mathit{k}}_3& 0\\ \multicolumn{1}{c}{0}& 0& 0& {\mathit{d}}_4·{\mathit{k}}_4\end{array}\right]}_{4\times 4}}\\ \multicolumn{1}{c}{=\left[\begin{array}{cc}\hfill [{\mathit{B}}_{11}{]}_{3\times 3}\hfill & [0{]}_{3\times 1}\\ \multicolumn{1}{c}{[0{]}_{1\times 3}}& {\mathit{d}}_4·{\mathit{k}}_4\end{array}\right]}\\ \multicolumn{1}{c}{{\mathit{K}}_1={\left[\begin{array}{ccc}\hfill k_{1z}\hfill & s_{1x}& s_{1y}\\ \multicolumn{1}{c}{k_{2z}}& s_{2x}& s_{2y}\\ \multicolumn{1}{c}{k_{3z}}& s_{3x}& s_{3y}\\ \multicolumn{1}{c}{k_{4z}}& s_{4x}& s_{4y}\end{array}\right]}_{4\times 3}=\left[\begin{array}{c}\hfill [{\mathit{K}}_{11}{]}_{3\times 3}\hfill \\ \multicolumn{1}{c}{[{\mathit{K}}_{12}{]}_{1\times 3}}\end{array}\right],}\\ \multicolumn{1}{c}{{\mathit{K}}_2={\left[\begin{array}{ccc}\hfill k_{1x}\hfill & k_{1y}& s_{1z}\\ \multicolumn{1}{c}{k_{2x}}& k_{2y}& s_{2z}\\ \multicolumn{1}{c}{k_{3x}}& k_{3y}& s_{3z}\\ \multicolumn{1}{c}{k_{4x}}& k_{4y}& s_{4z}\end{array}\right]}_{4\times 3}=\left[\begin{array}{c}\hfill [{\mathit{K}}_{21}{]}_{3\times 3}\hfill \\ \multicolumn{1}{c}{[{\mathit{K}}_{22}{]}_{1\times 3}}\end{array}\right]}\\ \multicolumn{1}{c}{{\mathit{K}}_3=\left[\begin{array}{ccc}\hfill u_{1z}\hfill & t_{1x}& t_{1y}\\ \multicolumn{1}{c}{u_{2z}}& t_{2x}& t_{2y}\\ \multicolumn{1}{c}{u_{3z}}& t_{3x}& t_{3y}\end{array}\right],{\mathit{K}}_4=-\left[\begin{array}{ccc}\hfill u_{1x}\hfill & u_{1y}& t_{1z}\\ \multicolumn{1}{c}{u_{2x}}& u_{2y}& t_{2z}\\ \multicolumn{1}{c}{u_{2z}}& u_{3y}& t_{3z}\end{array}\right]}\\ \multicolumn{1}{c}{{\mathit{s}}_i={\mathit{b}}_i\times {\mathit{k}}_i=[s_{\mathrm{ix}},s_{\mathrm{iy}},s_{\mathrm{iz}}{]}^T,{\mathit{t}}_i={\mathit{b}}_i\times {\mathit{u}}_i=[t_{\mathrm{ix}},t_{\mathrm{iy}},t_{\mathrm{iz}}{]}^T}\end{array}$$

Similarly, the Jacobian matrix of basic 3-PRS PM, which removes the UPS leg from the proposed PSH, can be obtained in singularity-free workspace

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{J}}_{\mathrm{au}}={\mathit{B}}_1^{-1}{\mathit{A}}_1}\\ \multicolumn{1}{c}{{\mathit{A}}_1={\mathit{K}}_{11}+{\mathit{K}}_{21}{\mathit{K}}_4^{-1}{\mathit{K}}_3,{\mathit{B}}_1={\mathit{B}}_{11}}\end{array}$$

(18)

Singularity-free orientation workspace

Singularity types

In the view of PM design, singularity is important for singularity-free path planning and for geometric design of a manipulator with singularity-free workspace. Many researchers in robotics community have focused on the issue of singularity for various purposes.^20,21 For the basic 3-PRS PM, manipulator singularity can be classified into four types, including inverse kinematic singularity (IKS), forward kinematic singularity (FKS), combined singularity, and constraint singularity. ⁸ The first three kinds of singularities can be obtained by analyzing rank deficiency of matrices ${\mathit{A}}_1$ and ${\mathit{B}}_1$ in equation (18) while the fourth kind of singularity can only be derived with the reciprocal screw theory. ²²

For the proposed PSH, the UPS leg is essential to make it pass through the singularity configurations of order one. As shown in Figure 5, when PRS leg1 is perpendicular to the corresponding sliding rail ${\mathit{A}}_1{\mathit{C}}_1$ , an IKS occurs with $det({\mathit{B}}_1)=0$ and the basic 3-PRS PM loses 1 DOF because an infinitesimal motion of the linear actuator ${\mathit{C}}_1^{\prime }$ will cause no motion of the moving platform. At this moment, the UPS leg turns active which helps the PSH regain 1 DOF and become 3-DOF again.

Figure 5.

Manipulator configuration when IKS happens to the basic 3-PRS PM: (a) CAD model (b) schematic diagram.

Comparing to the IKS, the avoidance of FKS is strongly demanded in view of hybrid machine tool design, because it will make the system uncontrollable and lead to severe accuracy degradation.

As known, the FKS occurs to the basic 3-PRS PM when A₁ is not full rank, for example, the PRS leg1 lies in the base plane ${\mathit{B}}_1{\mathit{B}}_2{\mathit{B}}_3$ with the extension of line ${\mathit{C}}_1{\mathit{B}}_1$ passing through the center point P of the moving platform, as shown in Figure 6. In this case, the moving platform gains one more DOF even when all the three actuators are locked. Fortunately, the troubles caused by FKS can also be resolved by the introduction of redundancy. In fact, when FKS happens, the UPS leg immediately turns active, which makes the PSH controllable again. In practice, the introduced redundant actuation can only help the PSH to eliminate FKS of one order.

Figure 6.

Manipulator configuration when FKS happens to the basic 3-PRS PM: (a) CAD model (b) schematic diagram.

The other two kinds of singularities, the combined singularity and the constraint singularity, are quite uncommon and can rarely happen. The former one can be avoided by careful structural design. Similarly, the latter one can rarely happen due to the physical constraints. Therefore, there will be no more further discussion about these two kinds of singularities in this article.

Singularity elimination in z-plane

According to the analysis above, occurrences of manipulator singularities are mainly correlated with the Jacobian matrix, which is strongly dependent on the architectural parameters and configurations of the target PM; thus, it is hardly impossible to describe the singularity distribution in analytical form; nevertheless, it is available to do this by numerically enumerating all optional configurations in the orientation workspace.

In this section, singularity distribution of the basic 3-PRS PM is studied without consideration of angle limitation of the S joints and the stroke limitation of the P joint. The boundary of the singularity configurations can be mapped with respect to the two independent Euler angles, γ and β, in a fixed plane parallel to the base plane ${\mathit{A}}_1{\mathit{A}}_2{\mathit{A}}_3$ . With the architectural parameters listed in Table 1, unconstrained singularity distribution for the basic 3-PRS PM in the fixed plane z₀ = 600 mm, can be derived by numerical searching, as shown in Figure 7.

Table 1.

Architectural parameters of the proposed PSH.

Parameters	Value (mm)
Diameters of the fixed base, $r_a$	262.5
Diameters of the moving platform, $r_b$	144.5
Length of the PRS leg, L	425.0
Offset distance of the R joint, $\left|h_i\right|$	37.5
Offset distance B4P, $r_m$	82.2

Figure 7.

Unconstrained singularity distribution at z₀ = 600 mm of (a) the basic 3-PRS PM and (b) the proposed PSH.

As depicted in Figure 7(a), it is can be observed that the two kinds of singularities, IKS and FKS, symmetrically distribute about $\gamma =0$ , while IKS-typed singularities always occur at configurations with large yaw and pitch angles. The desired orientation workspace of the proposed PSH is within the area A marked with dotted lines in Figure 7(a). It can be seen that for the basic 3-PRS PM, there is no IKS-typed singularities within the achievable orientation workspace. Referring to Figure 7(b), most singularity orientations depicted in Figure 7(a) are vanished, which exactly proves the validity of the introduction of the redundant leg.

Orientation workspace generation

In this section, we focus on studying the characteristic changes in the orientation workspace of the proposed PSH. Compared with serial ones, parallel manipulators generally have relatively small orientation workspace, which largely limits the applications of the parallel manipulators. In previous studies, the reachable orientation workspace is mainly generated under physical constraints imposed by the cone angle limits of the S joints, motion ranges of the linear actuators and interferences between the kinematic legs, whereas manipulator singularities are normally neglected. However, in practice, it is more meaningful for a PM to have an inconsecutive orientation workspace of singularity-free, in which the target PM is surely controllable.

Assuming that the cone angle, angle ψ in Figure 3, is within [−120°, 120°], the motion range of each linear actuator is within [0, 500 mm], and the allowable minimum distance between kinematic legs is set as 0.1 mm, the orientation workspace of the basic 3-PRS PM can be derived with the structure parameters listed in Table 1, as shown in Figure 8.

Figure 8.

Orientation workspace of the basic 3-PRS PM: (a) solid model and (b) singularity surface of FKS.

It can be seen from Figure 8 that with the modified S joints, two tilting angles, the yaw angle and the pitch angle, can be maximally enlarged to 100°, which will greatly enhance the applications of the basic 3-PRS PM. Cross sections in the middle range of the orientation workspace, which can be observed from the top view, are plotted in Figure 9.

Figure 9.

Cross sections of the orientation workspace at z₀ = 10, 60, 100, 370, 440, and 480 mm.

The singularity boundaries are highlighted with red asterisks in Figure 9. It is shown that singularities are still unavoidable problems to the basic 3-PRS PM and the orientation workspace is inconsecutive because of the singularity surface formed by FKS. However, the proposed PSH can achieve a singularity-free orientation workspace by virtue of kinematic redundancy. As shown in Figure 10, because of the introduced redundancy, singularity surfaces have all been eliminated and a thorough consecutive orientation workspace can be obtained.

Figure 10.

Singularity-free workspace of the proposed PSH and cross sections at z₀ = 100 and 370 mm.

Manipulability analysis

In previous researches, different measures have been devised to quantitatively evaluate the manipulability of a manipulator. One of the mostly commonly used measures is called manipulability index, ²³ which can be defined as follows

$$w=\sqrt{det\left({\mathit{J}}_a^T{\mathit{J}}_a\right)}$$

(19)

where ${\mathit{J}}_a$ is the Jacobian matrix. Generally, manipulability index can be used to search singularity configurations where the index values decrease sharply to 0. Implementing the respective Jacobian matrices of basic 3-PRS PM and the PSH, the value of manipulability indices can be obtained to evaluate and the influence of redundant leg will be discovered. With the structure parameters in Table 1, manipulability indices of the basic 3-PRS PM and the proposed PSH in horizontal plane z₀ = 100 mm are plotted in Figures 11 and 12, respectively.

Figure 11.

Manipulability indices of the basic 3-PRS PM at z₀ = 100 mm: (a) axonometric view and (b) left view.

Figure 12.

Manipulability indices of the PSH at z₀ = 100 mm: (a) axonometric view and (b) left view.

As shown in Figures 11 and 13(a), low manipulability indices values arise near singularity configurations depicted in Figure 9. Around the original point, the 3-PRS PM has relatively large manipulability indices which continuously decrease as the tilting angles increasing. The consecutive singularity-free orientation workspace is restricted within the central area around the original point. Compared with the basic 3-PRS PM, as shown in Figures 12 and 13(b), due to eliminating the singularity by introducing redundant leg, the manipulability index values of the PSH are all above 0.01 and the consecutive singularity-free orientation workspace is greatly enlarged.

Figure 13.

Top view of the distribution of the manipulability indices at z₀ = 100 mm: (a) the basic 3-PRS PM and (b) the proposed PSH.

The distribution of manipulability increments within the desired orientation workspace are calculated and depicted in Figure 14. It is can be observed that after introducing the redundant leg, the configurations, where the largest increment of manipulability index values occurs, are located along the singularity boundaries of basic 3-PRS PM, because the redundant leg eliminates those singularity configurations effectively. Meanwhile, it also less improves the manipulability in other poses. As a result, the overall manipulability of the PSH will be developed, especially at the singularity configurations, and its corresponding controllability is also improved.

Figure 14.

Manipulability improvements in the desired orientation workspace at z₀ = 100 mm: (a) axonometric view and (b) top view.

Prototype and experiments

Description of the prototype system

Based on the preceding analysis, a prototype system of the proposed PSH has been fabricated and assembled, as shown in Figure 15. Four AC motors (SGMAH-08ADA, from YASKAWA, Ltd., Japan) are employed as the main actuators to drive four groups of lead screw-nuts (R16-5T3-FS195, from HIWIN Co., Taiwan), respectively. Three groups of lead screw-nut components, which are vertically mounted along with three linear guides (from THK Ltd., Japan), are utilized as active P joints of the three PRS legs, while another group of lead screw-nut are used as active P joint of the UPS leg. The nominal motion stroke of each linear guide is about 1000 mm. Four modified S joints are mounted between the moving platform and the four kinematic legs. The 3-DOF motion of the PSH can be realized via the active motion of the four P joints and the passive motion of the R and S joints.

Figure 15.

Prototype of the proposed PSH.

The prototype system is controlled by an open architecture CNC system which includes a TURBO PMAC PCI motion controller (from Delta Tau Data System, Inc., USA), four servo drivers (SGDV-R90A01B, from YASKAWA Co., Ltd., Japan), an industrial PC (IPC-610-L, from ADVANTECH, Ltd., Taiwan), and so on. The four AC motors are controlled by four servo drivers, respectively, which are electrically connected to the TURBO PMAC PCI motion controller installed in the industrial PC. The industrial PC is used for motion control and man–machine interaction in the prototype system.

Preliminary experimental study

In this section, several preliminary experiments have been performed based on the proposed PSH, which has been theoretically proved to be capable of realizing both vertical and horizontal machining when used as a hybrid machine tool. A preliminary experiment is shown in Figure 16, which illustrates the superiority of the PSH over the basic 3-PRS PM. As depicted in Figure 16, the basic 3-PRS PM can swivel a maximum angle of about 61° about the y₀-axis before the prototype system becoming uncontrollable. According to the previous analysis, the trend of the manipulability indices in the experiment can be expressed in curves as shown in Figure 17, which illustrates that a FKS has arisen when the pitch angle is near 61° and the manipulability index of the PM is approaching 0. On the contrary, the proposed PSH can achieve a tilting angle of 90°, as shown in Figure 16, when the redundant leg is mounted. Meanwhile, the manipulability index values, as shown in Figure 17, are wholly promoted.

Figure 16.

Maximum swivel angle about y₀-axis: (a) the basic 3-PRS PM and (b) the PSH.

Figure 17.

Manipulability index curves in the preliminary experiment.

Furthermore, preliminary experiments have been performed to validate the tilting capacity of the proposed PSH, as shown in Figure 18, the moving platform is demanded to turn 90° around the x₀-axis and y₀-axis, respectively. Instantaneous postures are recorded at four interval tilting angles, which are 0°, 30°, 60°, and 90°, which shows that the proposed PSH can be smoothly operated under required instructions.

Figure 18.

Realization of horizontal-vertical conversion about (a) x₀-axis and (b) y₀-axis.

Conclusion and future works

1. To overcome the orientation workspace constraint of traditional PMs, a 3-DOF parallel swivel head with large tilting capability has been developed in this article, and a modified spherical joint with a maximum tilting angle of 120° is devised.

2. The kinematics of the PSH is analyzed and its Jacobian matrix is derived. Meanwhile, the kinematic redundancy is introduced in order to diminish the undesired orientation workspace that occupied by ill-conditioned configurations. Some detailed singularity analyses are established and the feasibility of eliminating singularity by introducing a redundant leg is verified.

3. The singularity-free orientation workspace of the PSH is generated based on the above theoretical analyses, whose volume is significantly bigger than that of traditional 3-PRS PM. Moreover, the manipulability of the PSH is also evaluated, which is essential for task workspace designing and singularity-free path planning. The result presents that the PSH obtains better manipulability, since the redundant leg mostly solves the singularity and improves the manipulability.

4. Experimental studies are performed using a fabricated prototype of the PSH and the results illustrate the feasibility of implementing the proposed PSH as a swivel head for a hybrid machine tool, which is expected to be capable of realizing horizontal and vertical machining at the same time.

5. Because the weak point of spherical joints still exists, which is the inevitable clearance due to the machining and assembling errors, the accuracy of the PSH should be further improved. Our future works focus on the accuracy improvement and static analyses of modified spherical joints, as well as other essential issues about the PSH, such as stiffness property, singularity-free path planning, error analysis, dynamics analysis, control strategy, and practical application.