Optimal design of a novel 4-degree-of-freedom parallel mechanism with flexible orientation capability

Abstract

This article proposes a novel spatial parallel kinematic mechanism with three kinematic limbs and 4 degrees of freedom. Its mobile platform can realize two translations and two rotation movements. Of note is that this mechanism can realize flexible A-/B-axis rotation (rotations about the x- and y-axes) and the transformation between vertical machining mode and horizontal machining mode. Using Tilt and Torsion angle description method, the kinematics analysis of this parallel kinematic mechanism is carried out. On this basis, the geometrical parameters, including parameters of three limbs and mobile platform, are optimized. In view of the particularity of configuration, the limb in the postposition with two active inputs is analyzed first and then the remaining two identical limbs. In the optimal design, motion/force transmissibility is taken into consideration, and the local transmission index is used as the performance evaluation criterion to investigate the orientation capability of this parallel kinematic mechanism. Using performance atlases method, the optimal kinematic design of the 4-degree-of-freedom parallel kinematic mechanism is carried out and a set of preferable optimized parameters is obtained at last. By virtue of the kinematic advantages, the proposed parallel kinematic mechanism can be used as the modular machining unit in five-axis machine centers. Based on this work, a machine tool with high flexibility has been developed to process the five faces of a workpiece within one setup.

Keywords

Parallel mechanism optimal design orientation capability motion/force transmissibility

Introduction

Due to the increasing complexity of workpieces, high-performance five-axis machining equipments have now been widely used in manufacturing industry and meanwhile have been studied intensively for better performance. In the research field of five-axis machining equipments, hybrid (serial–parallel) machine tools have drawn more and more researchers’ attention.^1,2 It is well known that the serial mechanisms usually have large workspace and simple kinematics but suffer from drawbacks like high inertia. In contrast, the parallel mechanisms have low inertia³ and compact structures but suffer from limited workspace⁴ and complex kinematics.⁵ By means of rational design and optimization, hybrid machining tools can gather the merits of serial and parallel mechanisms and avoid their negative effects. Based on this idea, some models have already been developed and successfully applied in industry, such as Tricept, Exechon and Ecospeed.^6–8

In some significant application fields, the workpieces are required to be high precision. So, installation accuracy, which has a great effect on machining accuracy and surface roughness, needs to be considered. One way to reduce the effect of installation on precision is to reduce the number of the installation process. So, five-face machining centers, which can machine five faces of the workpiece within one setup, avoid errors caused by multiple installations and have high practical value.⁹ The function of five-face machining makes demands for high orientation capability of machine tools, and it also requires that the machine tools can realize transformation between vertical machining mode and horizontal machining mode in five-face machining processes with only one setup. Therefore, in machine tool industry, there is an urgent need for high flexible orientation capability and large rotation range (at least 90° in order to realize the function of five-face machining). The existing solution adopted by most five-axis machining centers is to distribute the large rotation range to A-/B-axis rotary tables and swinging heads. In other words, the large rotational range is realized by the coordination movement of rotary tables and swinging heads. However, this method brings difficulties to control. Besides, the rotation range of rotary tables is normally small and the whole rotation range is usually less than 90°. So, new mechanisms are in great need to fulfill the goal in another way. Due to the previously presented advantages and good potentials, parallel kinematic mechanisms (PKMs) with flexible orientation capability and large rotational range provide a promising solution. This article is an attempt to propose such a PKM and deals with its kinematic optimization.

Compared with the 6-DoF fully parallel mechanisms, lower mobility parallel mechanisms, whose DoFs are less than six, have lower kinematics coupling and less interference among limbs.¹⁰ And in industry production, hybrid machining tools based on lower mobility parallel mechanisms (usually 3-DoF parallel tool heads) have been already applied.¹¹ Tricept hybrid robot is such a successful application, obtaining nearly 70% of the parallel equipments’ market.³ As Dr Neumann,¹² the inventor of Tricept, said, the success of this hybrid mechanism was owed to the 3UPS-UP (U: universal joint; P: prismatic joint; S: spherical joint) low-mobility parallel mechanism, which has fewer joints and less inefficient degrees of freedom. The reasons why the parallel mechanisms are hardly used in industry are none other than their complex structures and the resulting high kinematic, dynamic coupling and manufacturing difficulty.¹³ However, only few fully parallel 5-DoF mechanisms have been currently applied in industry production, such as series of fully parallel mechanisms developed by METROM.¹⁴ On the whole, the most popular parallel mechanisms in practice are those having simple joint configurations or (partly) symmetrical structures.

For the application of PKMs in manufacturing field, higher orientation capability and simpler kinematics are usually expected. Among the researches of hybrid/parallel mechanisms, one of the most important concerns is the high orientation capability.^15–17 Tricept machine tools have a serial A/C-axis (rotations about the x- and z-axes) tool head, leading to high rotational output and large workspace.¹⁸ However, the traditional serial A-/B-axis tool heads have drawbacks like complicated transmission system, huge structure, hard to install and hard to maintain. And in practical engineering application, serial architecture brings difficulties to locus planning and control because its two rotations are not coupled motions.¹⁹ Moreover, the finished surface is sometimes scratched by the rotating cutter during setup when using a serial articulated tool head.²⁰ By contrast, Ecospeed machining centers, developed by German DS-Technology,²⁰ can realize the function of the A-/B-axis rotation (rotations about the x- and y-axes) with a 3-PRS parallel module, which offers one translation degree of freedom and two rotational degrees of freedom, and avoids the drawbacks of serial A/C-tool head.²¹ Besides, the parallel machine module has an advantage in the flexibility of orientation over its serial counterpart. And flexible orientation capability offers the possibility of manufacturing complex freeform surfaces, such as surfaces of large-scale aircraft structure parts. For industry applications, the orientation output capability is supposed to be more than 40°.²⁰ In view of this, the orientation capability of the 4-DoF spatial parallel mechanism proposed in this article is investigated and it is also the main target to be optimized in order to get better performance in practical applications.

Sequentially, kinematic optimization with the goal of high orientation output should be carried out to derive the parameters of the discussed mechanism. As the precondition of dimension synthesis, performance evaluation is of vital importance, in which the performances to be evaluated and the method of executing evaluation process are the main concerns.²² Parallel manipulators have advantages in motion/force transmission. Thus, the performance index based on motion/force transmission is supposed to be taken into consideration in the design process of a certain PKM.²³ On the basis of the transmission angle, a local transmission index (LTI) was proposed in Wang et al.²⁴ to evaluate the motion/force transmissibility of PKMs. Using this performance evaluation index and its extending indices, the optimal kinematic design of the proposed 4-DoF PKM is carried out.

The remainder of this article is organized as follows. In section “Architecture description,” the proposed PKM is presented in detail and its mobility is analyzed. In section “Inverse kinematics analysis,” the inverse kinematics of this parallel mechanism is derived. Section “Optimal design” introduces the performance evaluation criterion which takes the orientation capability into consideration first. And on this basis, the optimization process is carried out, mainly considering its flexibility of A-/B-axis rotations and its capability to transform between vertical machining mode and horizontal machining mode. Section “Conclusion” concludes this article.

Architecture description

The CAD model and kinematic scheme of the novel 4-DoF parallel robot are shown in Figure 1. The parallel tool head has two translational and two rotational (2T2R) degrees of freedom. In order to gain high orientation capability, A-/B-axis rotational angles should be larger than 40°.²⁰ Besides, to machine the five faces of workpieces within one setup, the hybrid machine tool is supposed to realize the transformation between a vertical machining center and a horizontal one. To reach this goal, this article mainly investigates the orientation capability of this parallel module.

Figure 1.

A novel spatial 4-DoF parallel mechanism: (a) CAD model and (b) kinematic scheme.

The global frame $ℜ : O - XYZ$ is defined as follows. Plane O-XY is coplanar with the horizontal plane abc , the origin O is coincident with the base midpoint of the isosceles triangle abc ( $ac = bc$ ). The Y-axis is collinear with base ab and the Z-axis is perpendicular to plane abc . $ℜ' : o - xyz$ is a mobile frame which is fixed on the mobile platform. When the tool head is in its zero position, each axis in $ℜ'$ is parallel to its counterpart in $ℜ$ . Each of the vertexes of the triangular platform is connected to the base through a limb, which is referred to as the first ( B ₁ P ₁ ), the second ( B ₂ P ₂ ) and the third limb ( B ₃ P ₃ ). The first limb is in the same plane with the second one, and they have identical PRU (R: revolute joint) kinematic chains. The first and the second limbs are connected to the mobile platform through universal joints, and the other sides are connected to vertical moving slides through revolute joints. Of note is that the third limb in postposition has a different kinematic chain structure. It is connected to the platform through a spherical joint and connected to a horizontal slide through a revolute joint on the other side. Different from the former two slides, this horizontal slide is connected to a vertically derived slide. Thus, the third limb is also referred to be as PPRS chain structure. All slides are actuated by motors. In Cartesian coordinate system O-XYZ, the moving platform has four spatial DoFs, which are two translations along the Y- and the Z-axis and two rotations about the X- and the Y-axis. So, the whole mechanism can be referred to as 2T2R DoFs mechanism.

Except for offering an additional degree of freedom than normal 3-DoF head-tools, this mechanism can eliminate the influence of parasitic motion along the Y-axis. When unwanted motion along the Y-axis occurs with the rotation of platform, it can be compensated by the active motion of the platform.²⁵

As a result of two PRU kinematic limbs, the translation of the moving platform along the X-axis is restricted. So, there is no parasitic motion along the X-axis. As the third PPRS limb has six DoFs, the connection line P ₁ oP ₂ of the platform can move and rotate freely in the O-ZY plane. It is not hard to find that the two PRU limbs offer propulsion for the translation along Y-axis and rotation about X-axis, during which the PPRS limb offers driven movement due to platform geometry restriction. The propulsion for the rotation about Y-axis is provided by the third limb and the other two offer driven movements.

A potential application of this 4-DoF parallel mechanism is illustrated in Figure 2, which is a hybrid five-axis machining center. Apart from the parallel mechanism 3, this hybrid machining center has three serial parts, which are a moving slide along the x-axis 5, a moving slide along the y-axis 2 and a rotary table about the z-axis 4. The two slides, which make up a X-Y table, extend the stroke of parallel mechanism. The rotary table assists the realization of five-face machining.

Figure 2.

Hybrid machine tool with 4-DoF PKM: (1) workpiece, (2) moving slide along the y-axis, (3) proposed PKM, (4) rotary table about the z-axis and (5) moving slide along the x-axis.

This mechanism has large A-/B-axis rotation angle and can realize function of five-face machining. The implementation of five-face machining is like follows: when the moving platform is in its zero position, the machining center is in vertical machining mode and the upper surface of the workpiece 1 can be machined. After completion in upper surface machining, the moving platform rotates 90° about the y-axis and the machining center is in the horizontal machining mode. Cooperating with the rotary table, four vertical faces of the workpiece can be machined. In a word, the hybrid machining center can realize five-face machining in one setup and has a promising application prospect.

Inverse kinematics analysis

As shown in Figure 1(b), vertices of the platform are denoted as joints $P_{i}$ (i = 1, 2, 3), and the central points of revolute joints connected to the slides are denoted as $B_{i}$ (i = 1, 2, 3). Since the limbs $B_{1} P_{1}$ and $B_{2} P_{2}$ have the same PRU kinematics structure, the geometrical parameters of these two branches can be reasonably assumed to be the same for convenience. Then, the geometry parameters will be $| B_{1} P_{1} | = | B_{2} P_{2} |$ and $| o P_{1} | = | o P_{2} |$ . Since the rotational axis of the proposed mechanical configuration is parallel to the x-axis of the global frame $O - XYZ$ in its initial position and coincident with the y-axis of the mobile frame $o - xyz$ , the description method of the roll pitch and yaw (RPY) angles can be accepted for the orientation description. Under this method, the rotation matrix can be described as

\begin{matrix} R (ω, ξ, σ) = R_{x} (ω) R_{y} (ξ) R_{z} (σ) \\ = (\begin{matrix} \cos ξ \cos σ & - \cos ξ \sin σ & \sin ξ \\ \sin σ \cos ω + \sin ω \sin ξ \cos σ & \cos ω \cos σ - \sin ω \sin ξ \sin σ & - \cos ξ \sin ω \\ \sin ω \sin σ - \cos ω \sin ξ \cos σ & \sin ω \cos σ + \cos ω \sin ξ \sin σ & \cos ω \cos ξ \end{matrix}) \end{matrix} (1)

(1)

As the description in section “Architecture description,”P₁ and P₂ are constrained within the $o - yz$ plane. Thus, the x-components of the points P₁ and P₂ should be zero. Then, $\cos ξ \sin σ = 0$ can be obtained. While $ξ < 90 \circ$ , $σ = 0$ is identified. Consequently, the new rotation matrix is written as

R (ω, ξ, 0) = R_{x} (ω) R_{y} (ξ) R_{z} (0) = (\begin{matrix} \cos ξ & 0 & \sin ξ \\ \sin ω \sin ξ & \cos ω & - \cos ξ \sin ω \\ - \cos ω \sin ξ & \sin ω & \cos ω \cos ξ \end{matrix})

(2)

The geometrical parameters are denoted as follows, $B_{i} P_{i} = L_{i} (i = 1, 2, 3)$ , $P_{1} P_{2} = 2 D_{1}$ , $ab = 2 D_{2}$ , $o P_{3} = R_{1}$ and $Oc = R_{2}$ . And $(x, y, z)^{T}$ represents the position of the central point of moving platform in the global frame. For convenience, the methods of Tilt and Torsion (T&T) angles can be adopted.²¹ The orientation of moving platform can be described by the complement of tilt angle $φ$ and azimuth angle $θ$ . The transformation formats of RPY-angle and T&T-angle are shown as follows

ω = \sin^{- 1} (- \sin θ \sin φ)

(3)

ξ = \sin^{- 1} (\frac{\sin θ \cos φ}{\cos ω})

(4)

In all, parameters $(φ, θ, z, y)$ are used to describe the position and orientation of moving platform. In the slave coordinate system $o - xyz$ , the coordinates of $P_{i} (i = 1, 2, 3)$ can be expressed by

p_{1}^{'} = {[0 - D_{1} 0]}^{T}

(5)

p_{2}^{'} = {[0 D_{1} 0]}^{T}

(6)

p_{3}^{'} = {[- R_{1} 0 0]}^{T}

(7)

Multiplied by the coordinate rotation matrix $R (ω, ξ, 0)$ , the coordinates of $P_{i} (i = 1, 2, 3)$ can be derived under the global frame as follows

p_{i} = {[x_{i} y_{i} z_{i}]}^{T} = R (ω, ξ, 0) \cdot p_{i}^{'} + c (i = 1, 2, 3)

(8)

where by definition, $c = [x, y, z]^{T}$ is the position coordinates of moving platform center point under the global frame.

The coordinates of revolute joints centers under global frame are

b_{1} = {[0 - D_{2} ρ_{1}]}^{T}

(9)

b_{2} = {[0 D_{2} ρ_{2}]}^{T}

(10)

b_{3} = {[- R_{2} ρ_{3} ρ_{4}]}^{T}

(11)

where by definition, $ρ_{i} (i = 1, 2, 3, 4)$ is the z-axis coordinate of each actuated slide.

The constraint in the third branch denotes that slide B₃ and joint P₃ have the same y-axis coordinate. Besides, the mechanism offers the constraints $| P_{i} - B_{i} | = L_{i}$ . And the variables $ρ_{i}$ can be derived as

ρ_{1} = z - D_{1} \sin ω \pm \sqrt{L_{1}^{2} - {(y - D_{1} \cos ω + D_{2})}^{2}}

(12)

ρ_{2} = z + D_{1} \sin ω \pm \sqrt{L_{2}^{2} - {(y + D_{1} \cos ω - D_{2})}^{2}}

(13)

ρ_{3} = y - R_{1} \sin ω \sin ξ

(14)

ρ_{4} = z + R_{1} \cos ω \sin ξ \pm \sqrt{L_{3}^{2} - {(R_{2} - R_{1} \cos ξ)}^{2}}

(15)

For the presented configuration of the mechanism in Figure 1, the sign “+” in equations (12)–(15) is taken here.

Optimal design

In the optimal design, flexible orientation capability is the general objective. It is further reflected in two aspects: capacity for flexible A-/B-axis rotation and the ability to realize the transformation between vertical machining mode and horizontal machining mode. The performance-chart-based optimal design method²⁶ is used here since it provides all possible solutions instead of single one.

In the optimization process, each limb of the 4-DoF parallel mechanism has three parameters to be considered. They are the length of the limb, the distance between the central point of the moving platform and the contact point and the distance between the central point and the vertical slide axis. For PPRS limb, geometrical parameters to be optimized are $R_{1}, R_{2} and L_{3}$ . For the two identical PRU limbs, geometrical parameters to be optimized are $D_{1}, D_{2} and L$ .

To gain more reasonable results, the optimization process is divided into two parts. The PPRS limb in postposition is optimized first due to its special configuration, and then, the geometrical parameter optimization of the other two PRU limbs is carried out with the parameters derived from the first step.

The distributed process has the advantage of considering the orientation capabilities of different limbs over that assuming all limbs have identical parameters. Before the kinematic optimization design, geometric parameters should be normalized. Since the six parameters $R_{1}, R_{2}, L_{3}$ , $D_{1}, D_{2} and L$ are divided into two groups and optimized in two steps. The normalization of geometric parameters $(R_{1}, R_{2}, L_{3})$ is carried out as an example and the normalization of $D_{1}, D_{2}, L$ is carried out similarly.

Let $D = (R_{1} + R_{2} + L_{3}) / 3$ , and then, the normalized parameters can be derived as $r_{1} = R_{1} / D$ , $r_{2} = R_{2} / D$ and $l_{3} = L_{3} / D$ . According to the assembly configuration of the discussed mechanisms, the conditions $r_{1} + l_{3} \geq r_{2}$ and $r_{1} \leq r_{2}$ can be derived. By virtue of these conditions, an area as the hatched part can be identified. This area is referred to be as parameter design space and can also be expressed in rectangular coordinates $(s, t)$ as shown in Figure 3.

Figure 3.

Parameter design space: (a) spatial description and (b) planar description.

The mapping functions between $(s, t)$ and $(r_{1}, r_{2}, l_{3})$ can be derived as

s = l_{3}

(16)

t = \sqrt{3} - \frac{\sqrt{3}}{3} r_{1} - \frac{2 \sqrt{3}}{3} r_{1}

(17)

l_{3} = s

(18)

r_{1} = \frac{3 - s - \sqrt{3} t}{2}

(19)

r_{1} = 3 - r_{1} - s

(20)

In order to investigate the orientation capability of the proposed mechanisms, the motion/force transmission performance is taken into consideration in the optimization process. With screw theory¹⁹ as the mathematic foundation, the orientation capability investigation and optimal design process based on the existing Local Transmission Index²³ are carried out. And the results are shown using performance chart–based design methodology.¹⁹

Orientation capability analysis and optimal design of the PPRS limb

The separate optimization of the postposition, PPRS limb is equal to the optimization of a spatial four-bar mechanism with two inputs and two outputs. The kinematic scheme is shown in Figure 4. In view of practical requirements, the individual limb is supposed to realize flexible A-/B-axis rotation and the maximum of rotation about the y-axis needs to be more than 90°.

Figure 4.

Equivalent model of the PPRS limb.

The parameters used to describe the orientation of this spatial four-bar mechanism are $(ω, ξ)$ . Because the moving axes of the three actuators are parallel to the Z-axis, the performance of mechanism is identical in an arbitrary Z-axis position. Thus, the influence of motion along Z-axis can be neglected. Besides, the rotation about the y-axis and the translation along the y-axis are not decoupling. The movement of the platform along the y-axis has influences on the orientation capability of the PPRS limb. So, in the process of orientation optimization process, the position parameter y should be considered.

The inverse kinematic analysis is similar to the results in section “Inverse kinematics analysis.” The position of the actuator B ₃ can be expressed as $b_{3} = {[- r_{2} ρ_{3} ρ_{4}]}^{T}$ , where

ρ_{3} = y - r_{1} \sin ω \sin ξ

(21)

ρ_{4} = z + R_{1} \cos ω \sin ξ + \sqrt{L_{3}^{2} - {(R_{2} - R_{1} \cos ξ)}^{2}}

(22)

The unit input twist screws of the two actuators can be expressed as

$_{I 1} = (0, 0, 0; 0, 0, 1)

(23)

$_{I 2} = (0, 0, 0; 0, 1, 0)

(24)

And the transmission wrench screws can be derived by the method in Sun et al.²³ as

$_{T 1} = (s_{1}; r_{1} \times s_{1})

(25)

$_{T 2} = (s_{2}; r_{1} \times s_{2})

(26)

where by definition, $s_{1}$ is $\bar{B_{3} P_{3}}$ , $s_{1}$ is $(0, 1, 0)$ , $r_{1}$ is $\bar{o P_{3}}$ and $| s_{1} | = 1$ .

And unit output twist screws $$_{O 1}$ and $$_{O 2}$ of the mobile platform can also be derived similarly.

According to the definitions, the input transmission index (ITI) of PPRS limb can be expressed as

ρ_{I 1} = \frac{$_{T 1} \circ $_{I 1}}{{| $_{T 1} \circ $_{I 1} |}_{max}}, ρ_{I 2} = \frac{$_{T 2} \circ $_{I 2}}{{| $_{T 2} \circ $_{I 2} |}_{max}}

(27)

And the output transmission index (OTI) of PPRS limb can be expressed as

ρ_{O 1} = \frac{$_{T 1} \circ $_{O 1}}{{| $_{T 1} \circ $_{O 1} |}_{max}}, ρ_{O 2} = \frac{$_{T 2} \circ $_{O 2}}{{| $_{T 2} \circ $_{O 2} |}_{max}}

(28)

Consequently, the local transmission index (LTI) of the PPRS limb can be defined as

κ = min {| ρ_{I 1} |, | ρ_{I 2} |, | ρ_{O 1} |, | ρ_{O 2} |}

(29)

Based on this index, a good transmission workspace (GTW) under the description of T&T angles $(φ, θ)$ can be identified by the constraint of $κ \geq \sin 40 \circ$ .²⁰ In the GTW, there exists a maximal circular region defined by $0 \leq φ \leq 2 π, 0 \leq θ \leq θ_{max}$ . Then, the $θ_{max}$ is used to define the good transmission orientation capability (GTOC), which can be denoted by

W_{GTOC} = θ_{max}

(30)

First, let $y = 0$ and constrain $κ \geq \sin 40 \circ$ , the analysis of PPRS mechanism is carried out. For high A-/B-axis orientation flexibility, the constraint $W_{GTOC} \geq 40 \circ$ is set and the results are shown in Figure 5.

Figure 5.

Atlas of the $W_{GTOC}$ for the orientation of the PPRS limb.

In order to realize the transformation between a vertical machining mode and a horizontal machining mode, the rotation angle about the y-axis from the origin posture should be as far as 90° when $φ = 0$ . By constraining $κ \geq \sin 40 \circ$ , the distribution of $θ_{min}$ in the parameter design space is shown in Figure 6.

Figure 6.

Atlas of the $θ_{max}$ for the orientation of the PPRS limb.

A combination of criteria $W_{GTOC} \geq 40 \circ$ and $| θ_{min} | \geq 90 \circ$ indicates that no set of parameters can satisfy both conditions (see Figure 7).

Figure 7.

Optimum region for the parameters of the PPRS limb (considering $W_{GTOC}$ and $| θ_{min} |$ ).

One solution is to reduce the constraint to $W_{GTOC} \geq 35 \circ$ . But this will lead to the reduction of orientation capability of the parallel mechanism. So, in the following optimization process of PPRS limb, the fixed attitude angle of the moving platform $β_{0}$ (see Figure 8) is taken as a variable to optimize in the aim of realizing $W_{GTOC} \geq 40 \circ$ . In a small range, the fixed attitude angle has a positive effect on the $W_{GTOC}$ . As shown in Figure 8, the larger the fixed attitude angle $β_{0}$ is, the larger the parameter design space which satisfies the constraint $W_{GTOC} \geq 40 \circ$ (the bottom right corner of each line) becomes. However, large fixed attitude angle will also weaken the minimum of the rotation angle about the y-axis, which is $θ_{min}$ . As it is shown in Figure 9, the increase in $β_{0}$ goes with the shrinkage of the mid region between one pair of line. Of note is that when $β_{0}$ is larger than 9°, there is no optimal region that $| θ_{min} | \geq 90 \circ$ . In other words, the parallel mechanism can no longer realize the transformation function. Moreover, in view of practical application restrictions such as interference avoidance, $β_{0}$ should not be too large. On this basis, $β_{0} = 5 \circ$ is chosen in order to get a suitable optimum region.

Figure 8.

Schematic diagram of the fixed attitude angle of the moving platform $β_{0}$ .

Figure 9.

Different optimum region meeting constraint $W_{GTOC} \geq 40 \circ$ with different $β_{0}$ .

With $β_{0} = 5 \circ$ , an optimum region (the shaded area) can be identified and shown in Figure 10. A new set of parameters is selected from the parameter design space, $r_{1} = 0.75$ , $l_{3} = 1.35$ and $r_{2} = 0.90$ . And the corresponding indices are $W_{GTOC} = 41.253 \circ$ , $| θ_{min} | \geq 90 \circ$ .

Figure 10.

Optimum region for the parameters of the PPRS limb with $β_{0} = 5 \circ$ (considering $W_{GTOC}$ and $| θ_{min} |$ ).

Orientation capability analysis and optimal design of the two PRU limbs

After optimization in the previous section, a reasonable set of geometrical parameters of PPRS limb has been derived. Based on this result, kinematic optimization of the whole mechanism is carried out.

The parameter normalization process of the two PRU limbs is the same as the postposition PPRS limb. Normalization of the geometric parameters of the other two PRU limbs $(D_{1}, D_{2}, L)$ is carried out similarly. The normalized parameters can be derived as $d_{1} = D_{1} / D$ and $d_{2} = D_{2} / D, l = L / D$ , in which $D = (R_{1} + R_{2} + L_{3}) / 3 = (D_{1} + D_{2} + L) / 3$ .

Besides GTOC, a new evaluation criterion named global transmission index (GTI) is used to investigate global motion/force transmission performance over the derived GTW. It is defined as

ζ_{GTI} = \frac{\int \int {}_{ϖ}κ d θ d φ}{\int \int {}_{ϖ}d θ d φ}

(31)

In view of mechanism symmetry, $| B_{1} P_{1} | = | B_{2} P_{2} |, | Oa | = | Ob |, | o B_{1} | = | o B_{2} |$ are set. Thus, the number of the parameters to be optimized is reduced to three. With fixed attitude angle of the moving platform $β_{0} = 5 \circ$ and lateral displacement $y = 0$ , geometrical optimization is carried out similarly based on the evaluation criterion of motion/force transmission performance.

On account of four translational actuators, there are four unit input twist screws in all, which can be expressed as

$_{I 1} = (0, 0, 0; 0, 0, 1)

(32)

$_{I 2} = (0, 0, 0; 0, 0, 1)

(33)

$_{I 3} = (0, 0, 0; 0, 1, 0)

(34)

$_{I 4} = (0, 0, 0; 0, 0, 1)

(35)

The transmission wrench screws $$_{T i}$ and unit output twist screws $$_{O i}$ $(i = 1, 2, 3, 4)$ can be derived on the basis of methods in Li et al.²⁵ The descriptions of ITI $ρ_{I i}$ and OTI $ρ_{O i}$ of the 4-DoF mechanism are written as follows

ρ_{I i} = \frac{$_{T i} \circ $_{I i}}{{| $_{T i} \circ $_{I i} |}_{max}}

(36)

ρ_{O i} = \frac{$_{T i} \circ $_{O i}}{{| $_{T i} \circ $_{O i} |}_{max}}

(37)

The criterion LTI $κ$ of mechanism over all is

κ = min {| ρ_{I 1} |, | ρ_{O 1} |, | ρ_{I 2} |, | ρ_{O 2} |, | ρ_{I 3} |, | ρ_{O 3} |, | ρ_{I 4} |, | ρ_{O 4} |}

(38)

Based on the definitions, the performance atlases of $W_{GTOC}$ and $ζ_{GTI}$ can be derived and shown in Figures 11 and 12.

Figure 11.

Atlas of the $W_{GTOC}$ for the orientation of the whole parallel mechanism.

Figure 12.

Atlas of the $Γ_{GTI}$ for the orientation of the whole parallel mechanism.

By constraining $W_{GTOC} \geq 40 \circ$ , $ζ_{GTI} \geq 0.9$ , the shaded region in Figure 13 is derived, where a set of dimensionless parameters of the limb can be selected as $d_{1} = 0.70, l = 1.40, d_{2} = 0.90$ .

Figure 13.

Optimum region for the parameters of whole parallel mechanism (considering $W_{GTOC}$ and $Γ_{GTI}$ ).

So far, all normalized geometrical parameters of each limb have been derived: $| o P_{3} | = 0.75$ , $| OC | = 0.9$ , $| B_{3} P_{3} | = 1.35$ , $| Oa | = | Ob | = 0.9$ , $| o P_{1} | = | o P_{2} | = 0.7$ , $| B_{1} P_{1} | = | B_{2} P_{2} | = 1.4$ and $β_{0} = 5 \circ$ .

The provided geometric parameters are normalized and the orientation capability of the mechanism is only related to the ratio of this parameters. In practical application, the derived parameters should be multiplied by a scale factor D to derive the actual parameters, and the factor D is determined by the specific application requirements, namely, stroke. Take a potential application as an example: the scale factor D is set as 1000 mm, and the geometrical parameters with dimension are derived as $| o P_{3} | = 750 mm$ , $| OC | = 900 mm$ , $| B_{3} P_{3} | = 1350 mm$ , $| Oa | = | Ob | = 900 mm$ , $| o P_{1} | = | o P_{2} | = 700 mm$ , $| B_{1} P_{1} | = | B_{2} P_{2} | = 1400 mm$ and $β_{0} = 5 \circ$ .

On this basis, the GTW is investigated. With the lateral displacement $y = 0$ , the workspace is symmetrical and flexible A-/B-axis rotation can be realized (shown in Figure 14(a)). The maximum inscribed circle, expressed in red dashed line, denotes $W_{GTOC} = 41.3475 \circ$ by its definition. Thus, the PKM can realize flexible A-/B-axis rotation and the angle range is over $40 \circ$ . Along the 180°radial direction, the rotation angle about y-axis can reach $90 \circ$ , which means the transformation function between vertical machining mode and horizontal machining mode is obtained. However, since the lateral displacement y and the two orientation capability indexes, $W_{GTOC}$ and $ζ_{GTI}$ , have negative correlative, GTW under the condition $y = 0.3 \times d_{2}$ is investigated (see Figure 14(b)). When the off-center moving platform rotates, the GTW becomes asymmetry. And the maximum inscribed circle shrinks, which means $W_{GTOC}$ decreases. With $y = 0.3 \times d_{2}$ , current performance indices are $W_{GTOC} = 32.0856 \circ$ and $ζ_{GTI} = 0.8888$ , which are still acceptable. As for the transformation function between two modes, it can be seen that wherever the moving platform along the Y-axis is, target condition $| θ_{min} | \geq 90 \circ$ is always met, which means the transformation can still be realized. In short, the optimal geometric parameters are qualified and enable the whole PKM realize the function mentioned before.

Figure 14.

Good transmission workspace when $\sin 40^{\circ} \leq κ \leq 1$ : (a) $y = 0$ and (b) $y = 0.3 \times d_{2}$ .

Of note is that transformation function (i.e. vertical to horizontal) of this parallel mechanism is a big restriction on the parameter selection range in the design process. As shown in Figure 15, the constraint $| θ_{min} | \geq 90 \circ$ restricts the selection range of $β_{0}$ . If the flexible A-/B-axis rotation (large $W_{GTOC}$ and high $ζ_{GTI}$ ) is the only optimization objective, $β_{0}$ can be selected as large as $10 \circ$ . And the optimization result is shown in Figure 16. A set of parameters can be selected as follows $| Oa | = | Ob | = | OC | = 0.6$ , $| o P_{1} | = | o P_{2} | = | o P_{3} | = 0.4, | B_{1} P_{1} | = | B_{2} P_{2} | = | B_{3} P_{3} | = 2$ and $β_{0} = 10 \circ$ . Viewed from the GTW shown in Figure 17, the PKM with this set of parameters has large range of flexible A-/B-axis rotation angle ( $W_{GTOC} = 46.98 \circ, ζ_{GTI} = 0.9052$ ), but can no longer realize the transformation between vertical machining mode and horizontal machining mode. Thus, a conclusion can be drawn that transformation function is realized at the cost of a decrease in A-/B-axis rotation flexibility in the design process.

Figure 15.

Different optimum region meeting constraint $| θ_{min} | \geq 90^{\circ}$ with different $β_{0}$ .

Figure 16.

Optimum region when the constraint is only large $W_{GTOC}$ and high $ζ_{GTI}$ .

Figure 17.

Good transmission workspace when the constraint is only large $W_{GTOC}$ and high $ζ_{GTI}$ .

Prototype development

Based on the optimal design results and the concept presented in Figure 2, a prototype as illustrated in Figure 18(a) and (b) has been developed. As previously analyzed, the parallel model can realize the vertical–horizontal transformation as shown in Figure 18(c). On this basis, cooperating with a slider and a rotary table with continuous rotation capability, the developed machine is capable of five-face machining with single setup procedure. This is very helpful to the improvement of machining accuracy and orientation flexibility. The machine developed here has promising prospect in the process of structural components with freeform surfaces.

Figure 18.

Developed prototype: (a) CAD model, (b) overview and (c) vertical–horizontal transformation.

Conclusion

In this article, a novel 4-DoF parallel tool head is proposed based on practical demands and a set of optimized geometrical parameters has been obtained by considering the motion/force transmission performance. The main goal of optimization is to realize the function of flexible A-/B-axis rotations and the transformation between vertical machining mode and horizontal machining mode. Instead of assuming that three limbs have identical geometrical parameters, the optimization process is carried out in two steps. The PPRS limb in postposition is optimized first aiming at achieving flexible orientation capability and a large rotation range. Then, the whole parallel mechanism is analyzed and optimized. Besides, with the orientation description method of T&T angles, GTOC is derived and the effect of lateral displacement of moving platform on orientation capability has also been investigated. Optimization result obtained in the last section enables this 4-DoF parallel mechanism to fulfill the optimization goals and a prototype with high flexibility has been developed based on the work presented in this article. By virtue of flexible orientation capability and transformation function, the novel machining center can machine five faces of workpieces within one setup. And high machining precision and high working efficiency are possible to be achieved.

Footnotes

Appendix 1 Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grants 51675290 and 51425501. The second author (F.G. Xie) received the support from the Alexander von Humboldt (AvH) Foundation.

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