Research on the Conditions and Optimization of 3-UPU Parallel Manipulators That Naturally Realize Three Translational Degrees of Freedom

Abstract

The characteristics of the degrees of freedom (DOFs) of the 3-UPU parallel mechanisms (PMs) are closely related to the orientations of the ending R joints fixed connected with the platforms. The orientations of the ending revolute (R) joints in the Tsai 3-UPU PM cannot naturally realize three translational DOFs. First, this paper gives and proves the geometric conditions of the ending R joint fixed to the platforms that can naturally ensure three translational DOFs of the 3-UPU PMs. Second, based on the common motion/force index and good transmission workspace volume of the 3-UPU translational PMs, the performance map of the PMs is drawn, and the corresponding optimization design area of the PMs is obtained. Finally, combined with the obtained optimization design area, in view of the differences in the constraint and stiffness performance of the 3-UPU translational PMs, the multi-objective optimization is carried out using the game theory algorithm, and the orientations of the ending R joints for the 3-UPU translational PMs with good constraint and stiffness performance are obtained.

1. Introduction

The 3-DOF PMs with pure translational DOFs have high research value because of their particular kinematical properties. Clavel [1] first proposed the Delta PM, which is a widely known example of pure translational PMs. Chung-Ching et al. [2] proposed the translational PMs with doubly planar limbs based on the Lie group algebraic properties of the displacement set. Shen et al. [3] studied a 3-DOF translational PM with partial motion decoupling and analytic direct kinematics and analyzed its conditions of the singular configurations. Niu et al. [4] studied the type synthesis method for three translational parallel mechanisms considering velocity/force anisotropy. Zhang et al. [5] proposed a new jerk-optimal force and motion synchronous planning method for the 3-DOF translational force-controlled end-effector based on a 3-P(UU)2 PM. Lin et al. [6] designed a translational PM that uses only revolute joints and studied its kinematics, workspace, and singularity. For this class of PMs, the analysis of the 3-UPU translational PMs has become an interesting topic in recent years. Tsai et al. [7,8,9] first proposed and studied the 3-UPU PM with pure translational DOFs. Gregorio et al. [10] analyzed the mobility of the 3-UPU PM assembled for obtaining a pure translation motion and presented the analytic form and geometric interpretation of the singularity. Han et al. [11] studied the causes of gross self-motions of a 3-UPU PM and found that this type of PM is extremely sensitive to clearances and manufacturing errors in the bearing-shaft assembly. Lu et al. [12] proposed an asymmetric 3-UPU translational PM. Hu et al. [13] compared this PM with Tsai 3-UPU PM considering the singular configuration, constrained force/torque situations, and stiffness. Bhutani et al. [14] discussed the design considerations and sensitivity analysis of a 3-UPU translational PM with excellent practical feasibility and developed the prototype of this PM. Gan et al. [15] presented a metamorphic PM which can switch its motion between pure translational and pure rotational motions and investigated the characteristics and singularity of the PM. Kong et al. [16] presented a systematic classification of 3-UPU translational PMs based on constraint singularity loci using Gröbner Cover. Hassani et al. [17] derived a general explicit implicit dynamic model of the translational 3-UPU PM based on the principle of virtual work. Tsai et al. [18] proposed analytical joint compliance models for the translational 3-UPU PM. Zhao et al. [19] studied the kinematics and statics of a 3-UPU PM in screw coordinates. Xu et al. [20] investigated a 3-UPU wave compensation platform based on improved sliding mode control.

Through the above research, many important characteristics associated with the 3-UPU translational PMs have been revealed. However, for Tsai 3-UPU translational PM, the ending R joints in each UPU leg can not naturally keep parallel. Therefore, Tsai 3-UPU PM cannot obtain three translational DOFs naturally [13]. The condition that naturally ensures three translational DOFs of the 3-UPU PMs is still valuable work.

The performance analysis of PMs is an important research aspect of mechanisms. Motion/force transmission and constraint indices are widely used in the performance evaluation of PMs. Wang et al. [21,22] proposed the input and output motion/force transmission indices and used them for some non-redundant PMs. Meng et al. [23] proposed the modified output transmission index for high-speed PMs with articulated platforms based on the concept of equivalent transmission wrench screws. Liu et al. [24] proposed the motion/force constraint index for evaluating the constraint performance of the PMs and analyzed the constraint singularity based on this index. Marlow et al. [25] proposed a dimensionless constraint transmission index for analyzing the internal singularities of PMs with planar closed-loop subchains. Based on the constraint transmission index, Hu et al. [26] carried out a comparative study on the constraint performance of some PMs with 2R1T DOFs. Song et al. [27] analyzed the singularity and dimensional synthesis based on the motion/force transmission and constraint indices for the 3-UPU PM without parasitic motions.

The stiffness index is another important measure of PM performance. In this aspect, Huang et al. [28] developed a decomposition of a spatial stiffness matrix based on the stiffness eigenscrew problem and investigated the relationship between the eigenstiffnesses, the eigenscrews, and the stiffness matrix. Raoofian et al. [29] defined four stiffness indexes by decomposing the stiffness matrix into translational and rotational stiffness matrices, which represent the maximum and minimum values of the load resistance force/moment, respectively. Portman [30] evaluated the stiffness problem of Gough–Stewart platform by using the eigenvalues and eigenvectors of the stiffness matrix. Wu et al. [31] evaluated the linear stiffness and angular stiffness of the mechanism in the overall workspace by identifying the principal direction of the stiffness matrix. Yan et al. [32] proposed a virtual work index based on strain energy, which treats the linear and angular stiffness together to evaluate the stiffness performance distribution of Delta PM under different external load directions.

The multi-objective optimization algorithm is an effective method for obtaining PMs with good performance. Zhang et al. [33] used the particle swarm algorithm to simultaneously optimize the dexterous stiffness and reachable workspace of a bio-inspired PM. By considering the design requirements and pick-and-place trajectory, Wu et al. [34] optimized the kinematic and dynamic performances of a parallel Schönflies-motion robot with a prescribed workspace based on a non-dominated sorting genetic algorithm. Yang et al. [35] used a multi-objective optimization game algorithm (MOOGA) to optimize the parameters of the 2UPR-RPU and 2UPR-2RPU PMs considering the volume of the workspace, motion/force transmission and stiffness performance.

For the above reasons, the conditions of the R joints fixed to the platforms of the 3-UPU PMs that can naturally realize three translational DOFs are proposed and proved in this paper. For the proposed 3-UPU PMs, the orientations of the ending R joints have no effects on the kinematics but produce different constraint torque situations. Based on this feature, these PMs have the same motion/force transmission performance but different constraints and stiffness performances. For the common performance of the PMs, the atlas method is used to optimize the structure size with good motion/force transmission performance. Based on the optimized size of platforms, considering the different constraint and stiffness performance of the PMs, a multi-objective optimization is carried out to solve the optimal orientations of the ending joints on the platforms.

2. The Conditions of 3-UPU PMs Naturally Realize Pure Three Translational DOFs

2.1. The Problem in Tsai 3-UPU PM

The general 3-UPU PM include a moving platform m, a base B, and three UPU-type legs r_i (i = 1, 2, 3) (see Figure 1). Here, m is an isosceles triangle with three vertices a_i (i = 1, 2, 3) and o as its center. B is an isosceles triangle with three vertices A_i and O as its center. B and m are similar to each other. The edges of B and m satisfy A₁A₂ = A₂A₃. In each UPU leg, there is one U joint at A_i, one P joint along r_i and one U joint at a_i. The U joint at A_i(a_i) includes an outer R joint R_i1(R_i4) which is fixed on B(m) and one inner R joint R_i2(R_i3) which is perpendicular to the P joint. R_i1 and R_i4 are the two ending revolute joints in the UPU-leg connected to the base and moving platform, respectively. The two inner R joints are kept parallel.

In general case, the relations of R_i1, R_i2, R_i3, and R_i4 in the UPU-leg are satisfied as following:

R_i1⊥R_i2, R_i1⊥r_i, R_i2||R_i3, R_i3⊥R_i4

(1a)

In fact, the DOF of the 3-UPU PMs largely depends on the orientations of R_i1 and R_i4 on B and m. According to different orientations of the ending R joints, the 3-UPU PMs can achieve 3T, 3R, 2R1T and 3T1R DoFs.

For Tsai 3-UPU PM (see Figure 2), the orientations of the ending revolute joints on the platforms make it not a mechanism that can naturally realize 3T DOFs. The orientations of the ending revolute joints on the two platforms for Tsai 3-UPU PM are satisfied as follows:

R₁₁||A₂A₃, R₂₁||A₁A₃, R₃₁||A₁A₂, R₁₄||a₂a₃, R₂₄||a₁a₃,R₃₄||a₁a₂

(1b)

In past research, the Tsai 3-UPU PM was regarded as 3-DOF PM with pure three translations and thus satisfies an important geometrical condition as follows [9]:

R_i1||R_i4(i = 1, 2, 3)

(1c)

In fact, since R_i1 and R_i4 are the first and the last revolute joints of the UPU-type leg, respectively, there is no natural parallel relationship between the two R joints. To keep R_i1||R_i4(i = 1, 2, 3) in initial assembly is a difficult work. If R_i1||R_i4(i = 1, 2, 3) does not hold, the pure translational characteristic of Tasi 3-UPU PM will disappear.

From Equation (1a), it is known that R_i1 and R_i4 must lie in the same plane for each leg. If R_i1||R_i4 is not valid at initial assembly, R_i1 and R_i4 must intersect at a point c_i in each leg in general case.

It is known from Equation (1a) that R_i1, R_i4 and r_i lie in the same plane G_i, which leads to

$$R_{i1}\subset G_i, R_{i4}\subset G_i$$

(2a)

c_i lies on R_i1 and R_i4, namely

$$c_i\in R_{i1}, c_i\in R_{i4}$$

(2b)

From Equation (1b), it is known that

$$R_{i1}\subset B, R_{i4}\subset m$$

(3a)

From Equations (2b) and (3a), it leads to

$$c_i\in B, c_i\in m$$

(3b)

It is known from Equation (3b) that c_i lies on the intersection line of B and m, which leads to

$$c_i\in c$$

(3c)

where c is the intersection line of B and m. It is known from Equation (3c) that c₁, c₂, and c₃ lie on line c in general case.

According to the geometric method for determining constraint forces/torques, in the general case (see Figure 3), each leg has a constraint force F_c1, F_c2, and F_c3 that passes through the point c_i and parallel with R_i2. Since the independent rotation axes and constraint forces of the PMs satisfy the intersection or parallel relationship [36], line c is generally a rotation axis of the Tsai 3-UPU PM in general case. Under this case, the Tsai 3-UPU PM has no three translational DOFs.

2.2. Geometrical Conditions for 3-UPU PM Naturally Realize Pure 3-Translational DOFs

For the above issue of Tsai 3-UPU PM, this section proposes novel 3-UPU PMs that can naturally realize three translational DOFs, as shown in Figure 4. R_i1(i = 1, 2, 3) in B and R_i4 in m have identical arrangements. In addition to satisfying Equation (1a), the orientations of the ending R joints on the two platforms of the proposed 3-UPU PMs satisfy

$$R_{i1}\parallel R_{j1}, R_{i4}\parallel R_{j4}, R_{i1}\nparallel R_{k1}, R_{i4}\nparallel R_{k4}, i\ne j\ne k$$

(4a)

In what follows, we will prove this conclusion. Without losing generality, when setting the ending R joints in the first and third leg satisfy Equation (4a), we can obtain:

$$R_{11}\parallel R_{31}, R_{14}\parallel R_{34}$$

(4b)

As shown in Figure 4, draw line L₁ which passes through A₁ and is parallel with R₃₂, and line L₂ which passes through a₁ and is parallel with R₃₃. Then, R₁₂ and L₁ can form plane M₁, and R₁₃ and L₂ can form plane M₂. L₁ and L₂ satisfy the following relation:

L₁||L₂

(4c)

For the 3-UPU PMs, in general case, $R_{12}\nparallel R_{32}, R_{13}\nparallel R_{33}$ are satisfied.

Combined with Equations (1a) and (4c), the relation of M₁ and M₂ can be obtained as following:

$$M_1\parallel M_2$$

(5a)

From Equations (1a) and (4b), it leads to

$$R_{11}\perp M_1, R_{31}\perp M_1, R_{14}\perp M_2, R_{34}\perp M_2$$

(5b)

From Equation (5a,b), it leads to

$$R_{11}\parallel R_{14}, R_{31}\parallel R_{34}$$

(5c)

For the second UPU leg, draw line l₁ which passes through A₂ and parallel with R₁₁, and line l₂ which passes through a₂ and parallel with R₁₄. Then, R₂₂ and l₁ can form a plane p₁, and R₂₃ and l₂ can form a plane p₂, as shown in Figure 5. l₁ and l₂ satisfy the following relation:

l₁||l₂

(6)

Let θ₁₁ be the angle between R₂₂ and l₁ and θ₂₁ be the angle between R₂₃ and l₂. Without loss generality, let θ₁₁ and θ₂₁ be two obtuse angles.

Let θ₁₂ be the angle between R₂₁ and l₁, and let θ₂₂ be the angle between R₂₄ and l₂. From the symmetrical structure of R_i1 in B and R_i4 in m, it is known that

θ₁₂ = θ₂₂

(7a)

Since R₂₂||R₂₃, l₁||l₂, there must be

θ₁₁ = θ₂₁

(7b)

p₁||p₂

(7c)

Let B₂ be a point on line R₂₁, line segment B₂C₂ is drawn which starts at B₂ and ends at C₂ on p₁; the following equation is satisfied

B₂C₂⊥p₁

(8a)

Let b₂ be a point on line R₂₄, line segment b₂c₂ is drawn which starts at b₂ and ends at c₂ on p₂; the following equation is satisfied

b₂c₂⊥p₂

(8b)

Then,

B₂C₂||b₂c₂

(8c)

Since R₂₂⊥R₂₁, R₂₃⊥R₂₄, are based on three perpendicular line theorem and converse theorem, there must be

R₂₂⊥A₂C₂, R₂₃⊥a₂c₂

(9a)

Considering R₂₂||R₂₃, there must be

A₂B₂C₂||a₂b₂c₂

(9b)

Since R₂₂⊥r₂, R₂₄⊥r₂, there must be

r₂ ∈ A₂B₂C₂, r₂ ∈ a₂b₂c₂

(9c)

The two parallel planes A₂B₂C₂ and a₂b₂c₂ have one common line which is coincident with r₂; thus, A₂B₂C₂ and a₂b₂c₂ are coplaner. Since R₂₁ and R₂₄ are coplaner, R₂₁ and R₂₄ are in plane C₂A₂a₂b₂. From Figure 5, it can be seen that

θ₁₁ = θ₁₃ + 90°, θ₂₁ = θ₂₃ + 90°

(10a)

Considering Equations (10a) and (7b), it leads to

θ₁₃ = θ₂₃

(10b)

Based on Equations (7a) and (10b) and the minimum angle theorem, it leads to

θ₁₄ = θ₂₄

(10c)

Thus, in plane C₂A₂a₂b₂, there must be

R₂₁||R₂₄

(11)

From the above analysis, it can be seen that when the conditions of Equation (4a) are satisfied, the PM will not have the situation as Tsai 3-UPU PM and can maintain three translations naturally.

Here, it should be noted that some references have been revealed that the SNU TPM has seven operation modes [11,12] and the Tsai TPM has five operation modes [13]. Many other PMs also have multiple operation modes. However, this paper does not provide a detailed analysis of the operation modes of PMs.

For the proposed 3-UPU PMs, although the orientation of the ending R joints on the platforms is different, these PMs have three translational DOFs and identical actuator layouts; thus, they have identical kinematics. However, the different orientations of the ending R joints produce different constraint torque situations [9,13] and thus produce different constraint and stiffness performances.

3. Optimization Design of the 3-DOF Translational 3-UPU PMs Based on Motion/Force Transmission Index

The proposed 3-UPU translational PMs have identical kinematics; therefore, they have identical motion/force transfer performance. This section will optimize the PMs from the aspect of the motion/force transmission index. In this paper, we only studied the cases where the ending R joints are located in the B and m since the assembly are relatively easy in these cases.

3.1. Motion/Force Transmission Performance Atlases of the 3-UPU Translational PMs

The input transmission performance index λ_Ii of the i-th limb is the power coefficient between $_Ii and $_Ti, and the minimum value of λ_Ii for all limbs is the input transmission performance index of the PMs [15], which is defined below.

$${\lambda }_{\mathrm{I}}=\min ({\lambda }_{ \mathrm{I}i})=\min ({\displaystyle \frac{\left|{{\displaystyle \mathit{＄}}}_{\mathrm{I}i}\circ {\mathit{＄}}_{\mathrm{T}i}\right|}{{\left|{\mathit{＄}}_{\mathrm{I}i}\circ {\mathit{＄}}_{\mathrm{T}i}\right|}_{\max }}}) ,(i = 1, 2, 3)$$

(12a)

For the 3-UPU translational PM, the input transmission performance index (ITI) is 1 [21].

The output twist screw $_Oi is the twist screw that exists at the moving platform after locking all the remaining actuated joints except the i-th limb. The power coefficient between $_Oi and the corresponding $_Ti is the output transmission performance indices λ_Oi of the i-th limb, and the minimum value of λ_Oi for all limbs is the output transmission performance index of the PMs [17], which is defined below.

$${\lambda }_{\mathrm{O}}=\min ({\lambda }_{\mathrm{O}i})=\min \left({\displaystyle \frac{\left|{\mathit{＄}}_{\mathrm{O}i}\circ {\mathit{＄}}_{\mathrm{T}i}\right|}{{\left|{\mathit{＄}}_{\mathrm{O}i}\circ {\mathit{＄}}_{\mathrm{T}i}\right|}_{\max }}}\right), (i=1,2,3)$$

(12b)

The minimum of λ_I and λ_O is used as a local transmission index to measure the motion/force transmission performance of the manipulators, which is defined below [17]:

$$\mathrm{LTI}=\min ({\lambda }_{\mathrm{I}}, {\lambda }_{\mathrm{O}})$$

(12c)

To evaluate the global transmission performance of the manipulators, the region with LTI ≥ 0.7 is defined as the good transmission workspace (GTW), and its volume is the good transmission workspace volume (GTWV). The global transmission index (GTI) [17,22] in GTW is defined as

$$\mathrm{GTI}={\displaystyle \frac{{\displaystyle \int \left(\mathrm{LTI}\right)dW}}{{\displaystyle \int dW}}}$$

(13)

where W denotes the region of the GTW of the manipulator.

Assume that the dimension parameters of the 3UPU PMs are given as A₁A₃ = 2E, a₁a₃ = 2e, the heights on the bottom edge of B and m are given as E₁ and e₁. Then, E/E₁ = e/e₁ is satisfied. The range of the actuators is set as 0.7E ≤ l_i ≤ 1.35E (i = 1, 2, 3). The size parameters of the PMs are set as: E = 0.4 m, E₁ = 0.8 m, e = 0.2 m, and e₁ = 0.4 m. The distributions for the LTI in the plane of Z = 0.7 m and its two-dimensional contour atlas are shown in Figure 6a,b, respectively.

3.2. Dimension Optimization Design Based on Motion/Force Transmission Index

In order to determine the dimension parameters of the 3-UPU PMs with good GTI and GTWV, a dimensional optimization design is performed. Since B and m are similar to each other, three are three of the four parameters (E, E₁, e, and e₁) need to be optimized.

To realize dimensional dimensions, the parameter design space is used to turn the geometric parameters into non-dimensional quantities [17,18]. The dimension parameters are normalized as D = (E + e + E₁)/3, r₁ = E/D, r₂ = e/D, r₃ = E₁/D, where D is a normalization factor. For the 3-UPU PMs, the conditions for the dimensionless quantities are 0 < r₁, r₂, r₃ < 3, r₂ ≤ r₁. The parameter design space and its planar expression are shown in Figure 7a,b, respectively.

The relationship among the normalized parameters can be expressed as:

$$\left\{\begin{array}{l}{r}_2=S\hfill \\ r_1={\displaystyle \frac{\sqrt{3}T-S}{2}}\hfill \\ r_3=3-r_1-r_2\hfill \end{array}\right.\mathrm{or} \left\{\begin{array}{c}S=r_2\\ T={\displaystyle \frac{3-r_3+r_1}{\sqrt{3}}}\end{array}\right.$$

(14)

Based on the GTI and the GTWV, the design atlases of the optimization indices are obtained, as shown in Figure 8. In Figure 8, the GTI and GTWV exist in the region surrounded by the blue line, the GTI decreases with the increase in S and T, while GTWV achieves a great value in the middle region surrounded by the blue line, and the value decreases along the surrounding direction. To make the PMs have better transmission performance and good transmission workspace volume, the indices GTI and GTWV should be as large as possible. Therefore, by setting GTI ≥ 0.8 and GTWV ≥ 0.8, an optimum area as the shadow region revealed in Figure 9 can be obtained.

After the optimal region was obtained, we selected a set of dimensionless parameters S = 0.36 and T = 1.5 from the optimal area, and D = 1.5 m was set, then the actual dimensions of the manipulators are e = 0.54 m, e₁ = 0.734 m, E = 1.6786 m, and E₁ = 2.2814 m. We set Z = 0.7, E = 1.175 m and drew the transmission atlas of the 3-UPU PMs with those optimized dimensions, as shown in Figure 10.

Comparing Figure 6 with Figure 10, it is clear that the LTI of the 3-UPU PMs is better than that before optimization, and the area with LTI ≥ 0.85 in the atlas is significantly increased. The GTI and GTWV values before and after optimization were considerably improved compared with the previous ones, as shown in

Table 1. Performance indices values of the 3-DOF translational 3-UPU PMs before and after optimization.

Index	e(m)	e1(m)	E(m)	E1(m)	GTI	GTWV
Before optimization	0.5	0.7	1.5	2	0.8159	0.5599
Optimized	0.5400	0.7340	1.6786	2.2814	0.8334	0.8657

.

4. Optimization of the Orientations of the Ending R Joints

In Section 3, the optimized dimensions of the 3-UPU translational PMs based on the identical motion/force transfer performance measured by GTI and GTWV are obtained. However, for the 3-UPU translational PMs with different orientations of the ending R joints, the constraint and stiffness performances are different. In order to find the orientations of the ending R joints having optimal constraint and stiffness performances, the multi-objective optimization is performed using the game theory algorithm in this section.

4.1. Constraint Performance Index

Let $_Ci be the constraint wrench screw in the i-th limb. The virtual output twist screw ∆$_Oi is the twist screw that exists at the moving platform after locking all actuators and releasing $_Ci [24,27]. $_Ci and ∆$_Oi can be calculated using the following formula:

$$\left\{\begin{array}{c}{＄}_{\mathrm{T}j}\circ \Delta {\$}_{\mathrm{O}i}=0\\ {＄}_{Ck}\circ \Delta {＄}_{\mathrm{O}i}=0\end{array}\right., j=1,2,3, k=1,2, \cdots ,k\ne i$$

(15a)

The output constraint index (OCI) of PMs is defined as

$$\mathrm{OCI}=\min \left({\displaystyle \frac{\left|{＄}_{\mathrm{c}i}\circ \Delta {＄}_{\mathrm{O}i}\right|}{{\left|{＄}_{\mathrm{c}i}\circ \Delta {＄}_{\mathrm{O}i}\right|}_{\max }}}\right),(i=1,2,3)$$

(15b)

For the above 3-UPU TPMs, the manipulators have different outer R joint layouts, so the constraint screw $_Ck is different, then the output constraint index OCI is different.

Similarly to GTW, the region of OCI ≥ 0.7 in the manipulator performance profile is defined as the good constraint workspace (GCW) and the global constraint index GCI of the manipulator within the GCW is defined as

$$\mathrm{GCI}={\displaystyle \frac{{\displaystyle \int \left(\mathrm{OCI}\right)dW}}{{\displaystyle \int dW}}}$$

(16)

where W denotes GCW.

4.2. Stiffness Performance Index

The stiffness model of Tsai 3-UPU translational PM considering active forces and constraint torques has been established in reference [13]. This stiffness model is also suitable for the proposed PMs in this paper. This model can be expressed as follows:

$$\left[\begin{array}{c}F\\ T\end{array}\right]=\mathbf{K}\left[\begin{array}{c}\delta p\\ \delta \psi \end{array}\right], \left[\begin{array}{c}\delta p\\ \delta \psi \end{array}\right]=\mathbf{C}\left[\begin{array}{c}F\\ T\end{array}\right], \mathbf{C}=\left[\begin{array}{cc}{\mathbf{C}}_{11}& {\mathbf{C}}_{12}\\ {\mathbf{C}}_{21}& {\mathbf{C}}_{22}\end{array}\right]$$

(17a)

where K is the stiffness matrix, C and C_ij are the compliance matrix and sub-compliance matrices of the PMs, respectively, F and T are the force and torque applied on the moving platform, respectively. δp and δψ are the corresponding linear deformation and angular deformation of the moving platform, respectively.

Let δp_F and δψ_F be the linear deformation and the angular deformation, respectively, produced by an arbitrary unit force F_I. Let δp_T and δψ_T be the linear deformation and the angular deformation, respectively, produced by an arbitrary unit torque T_I.

$$\left\{\begin{array}{c}\delta p_F={\mathbf{C}}_{11}{F}_I, \delta {\psi }_F={\mathbf{C}}_{21}{F}_I\\ \delta p_T={\mathbf{C}}_{12}{T}_I, \delta {\psi }_T={\mathbf{C}}_{22}{T}_I\end{array}\right.$$

(17b)

For the PMs, the smaller the maximum deformation is, the better the stiffness is 1 [34]. Thus, in this paper, the maximum of δp_F, δψ_F, δp_T, and δψ_T are used to evaluate the stiffness of the 3-UPU PMs [34], which can be calculated as follows:

$${‖\delta p_F‖}_{\max }=\sqrt{{\lambda }_{P_F\max }}, {‖\delta {\psi }_F‖}_{\max }=\sqrt{{\lambda }_{{\psi }_F\max }}, {‖\delta p_T‖}_{\max }=\sqrt{{\lambda }_{P_T\max }}, {‖\delta {\psi }_T‖}_{\max }=\sqrt{{\lambda }_{{\psi }_T\max }}$$

(17c)

where λ_PFmax, λ_ψFmax, λ_PTmax, and λ_ψTmax, are the maximum eigenvalues of ${\mathbf{C}}_{11}^T$C₁₁, ${\mathbf{C}}_{21}^T$C₂₁, ${\mathbf{C}}_{12}^T$C₁₂, and ${\mathbf{C}}_{22}^T$C₂₂, respectively.

The global linear stiffness index generated by F and T are expressed as

$$\left\{\begin{array}{c}{\eta }_{\delta P_{F\max }}={\displaystyle \frac{{\displaystyle {\int }_v{‖\delta p_F‖}_{\max }}}{{\displaystyle {\int }_{v}dV}}}, {\eta }_{\delta {\psi }_{F\max }}={\displaystyle \frac{{\displaystyle {\int }_v{‖\delta {\psi }_F‖}_{\max }}}{{\displaystyle {\int }_{v}dV}}}\\ {\eta }_{\delta P_{T\max }}={\displaystyle \frac{{\displaystyle {\int }_v{‖\delta p_T‖}_{\max }}}{{\displaystyle {\int }_{v}dV}}}, {\eta }_{\delta {\psi }_{T\max }}={\displaystyle \frac{{\displaystyle {\int }_v{‖\delta {\psi }_T‖}_{\max }}}{{\displaystyle {\int }_{v}dV}}}\end{array}\right.$$

(18)

where V is the total workspace volume.

4.3. Multi-Objective Optimization of the 3-UPU PMs

This paper uses the game theory algorithm to perform multi-objective optimization on the performance of the PMs. The details of multi-objective optimization using the game theory algorithm are not introduced here due to space limitations. More details are available in [35,37].

We set θ_i (i = 1, 2) to be the angle between R_i1(R_i4) and the A₁A₃(a₁a₃) axis in B(m), which varies from 0 to π, θ_i = [θ₁ θ₂]. In order to achieve a high constraint index, the corresponding objective functions are defined as

$${\mathit{f}}_1(\theta )=\mathrm{GCI}\to \max$$

(19a)

To maximize the stiffness performance, four objective functions are set as

$$\left\{\begin{array}{c}{\mathit{f}}_2(\theta )={\eta }_{\delta P_{F\max }}\\ {\mathit{f}}_3(\theta )={\eta }_{\delta P_{T\max }}\\ {\mathit{f}}_4(\theta )={\eta }_{\delta {\psi }_{F\max }}\\ {\mathit{f}}_5(\theta )={\eta }_{\delta {\psi }_{T\max }}\end{array}\right.\to \min$$

(19b)

Mathematically, the multi-objective optimization problem of the 3-UPU translational PMs can be expressed as

$$\begin{array}{l}\begin{array}{cc}\mathrm{maximize}& {\mathit{f}}_1(\theta )\end{array}\\ \begin{array}{cc}\mathrm{minimize}& {\mathit{f}}_2(\theta )\end{array}, {\mathit{f}}_3(\theta ), {\mathit{f}}_4(\theta ), {\mathit{f}}_5(\theta )\\ \begin{array}{cc}over& \theta =\left[{\theta }_1,{\theta }_2\right]\end{array}\\ {\mathrm{subject} \mathrm{tog}}_1:1.175m\le l_i\le 2.182m, i=1,2,3, {\mathrm{g}}_2:0^{\mathrm{o}}\le \left\{{\theta }_1,{\theta }_2\right\}\le {180}^{\mathrm{o}}\end{array}$$

(20)

Combined with the constraints given by Equation (20), the distribution diagrams of the objective function are plotted in the complete parameter space, as shown in Figure 11.

In multi-objective optimization using the game theory algorithm, the five objective functions are considered to have the same important status; thus, the weight factors of the five objective functions are all set as 1/5. Then, each objective is compromised simultaneously to obtain a balanced intersection subspace for all five. The step size of the angle variable parameter is set as 0.1°, namely, each objective is calculated for a total of 3,243,601 points in the complete objective parameter space.

After the objectives play multiple rounds according to their respective weights, the optimal balance point and the corresponding performance index values can be obtained as shown in

Table 2. The optimum parameter balance point and the corresponding performance index values.

Optimum Balance Point (θ1, θ2)	GCI	ηδPFmax	ηδPTmax	ηδψFmax	ηδψTmax
[97.5°, 3.7°]	0.7821	4.827 × 10−6	7.566 × 10−6	7.566 × 10−6	1.227 × 10−5

.

The results show that when the angle between R₁₁ and the A₁A₃ axis reaches 97.5°, and the angle between R₂₁ and the a₁a₃ axis reaches 3.7°, the global constraint index is 0.7821 and the global linear translation and rotation stiffness index generated by F and T are 4.827 × 10⁻⁶, 7.566 × 10⁻⁶, 7.566 × 10⁻⁶, and 1.227 × 10⁻⁵. The game stops at this point and all five performance indices have been optimized. At the optimal balance point, the five objective functions reach a balance position, but none of the five reach their maximum.

5. Conclusions

The contribution of this article lies in the proposing of the orientations of the ending R joints in the 3-UPU PMs that can naturally realize three translational DOFs and the optimization of them. It is proven that the geometric condition for the 3-UPU PMs to realize the natural three-translational motion is that the ending R joints of any two limbs are parallel. Compared to traditional 3UPU parallel mechanisms, the configuration schemes can naturally realize three translational DOFs and thus can be used in packaging, pick-and-place, assembly, and material handling.

The proposed 3-UPU PMs have identical motion/force transmission performance. The GTI and GTWV of the 3-UPU PMs are calculated and the optimum area with good GTI and GTWV is derived. The proposed 3-UPU PMs have different constraint and stiffness performances. Based on the multi-objective optimization and using MOOGA, the optimal orientation of the ending R joints of the 3-UPU translational PMs is obtained. This study realizes the configuration optimization of the PMs considering transmission performance, constraint performance and stiffness performance.