Type Synthesis of Fully Decoupled Three Translational Parallel Mechanism with Closed-Loop Units and High Stiffness

Abstract

In order to solve the problem of weak stiffness of the existing fully decoupled parallel mechanism, a new synthesis method of fully decoupled three translational (3T) parallel mechanisms (PMs) with closed-loop units and high stiffness is proposed based on screw theory. Firstly, a new criterion for the full decoupled of PMs is presented that the reciprocal product of the transmission wrench screw matrix and the output twist screw matrix of PMs is a diagonal matrix, and all elements on the main diagonal are nonzero constants. The forms of the transmission wrench screws are determined by the criterion. Secondly, the forms of the actuated and unactuated screws can be obtained according to their relationships with the transmission wrench screws. The basic decoupled limbs are generated by combination of the above actuated and unactuated screws. Finally, a closed-loop units construction method is investigated to apply the decoupled mechanisms in a better way on the high stiffness occasion. The closed-loop units are constructed in the basic decoupled limbs to generate a high-stiffness fully decoupled 3T PM. Kinematic and stiffness analyses show that the Jacobian matrix is a diagonal matrix, and the stiffness is obviously higher than that of the coupling mechanisms, which verifies the correctness of the proposed synthesis method. The mechanism synthesized by this method has a good application prospect in vehicle durability test platform.

1 Introduction

In recent years, the lower-mobility parallel mechanism (PM) has become an important issue in the field of robotics due to its advantages such as low cost, simple development of control system and high performance. It is widely used in precision machining [1], aerospace [2], micro-surgery [3], 3D printing [4], spatial positioning [5] and other fields. However, the control system of the coupled PM is relatively complex compared with the decoupled one because of the strong coupled relationship between the input and output of the coupled PM [6], and it is difficult to analyze the kinematics and dynamics as well as singular positions of the mechanism [7, 8]. The theoretical analysis and prototype development will be simpler if the full decoupled of the PM is realized, that is, the input and output variables represent a one-to-one correspondence relationship. Therefore, the decoupled PM has become one of the research hotspots, and many researchers have studied them to solve the problem of difficult control of the coupled mechanisms.

Based on topological structure design theory of position and orientation characteristic (POC) set, Liu et al. [9] proposed three methods for structure coupling-reducing of the 3UPS&UP PM based on the mechanism topology theory.

Briot et al. [10] presented a novel 4-DOF decoupled PM with Schoenflies motions, called Pantopteron-4. Zhang et al. [11] proposed a novel 3-DOF decoupled spatial PM. Jin et al. [12] proposed a 2R1T partially decoupled PM based on topological structure. Qu et al. [13] proposed a new method for type synthesis of decoupled PMs with low-mobility and redundancy constraints. Zhang et al. [14] synthesized a family of fully decoupled spherical mechanisms according to the screw theory. Jin et al. [15] revealed the relationship between kinematic decoupled and basic kinematic chains, proposed a structure synthesis principle of topological decoupled mechanisms, and synthesized partially decoupled 3-degree-of-freedom (DOF) PMs. Lin et al. [16] studied a new type of mechanism SLBMs with three branches, and the translational and rotational motions are partially decoupled. Gogu [17] utilized linear transformation theory to study the type synthesis of nonsingular and completely isotropic 2T2R PMs. Yang et al. [18] proposed a 3-RPRS 6-DOF decoupled PM in which the three planar motion DOFs and the other three motion DOFs are independently controlled. On the basis of the topological structure characteristics of PMs, Glazunov [19] applied screw theory to obtain several decoupled 3T PMs. Kong et al. [20] used geometric description method of forward kinematic analysis to study the translational decoupled PMs. Liu et al. [21] designed a class of two to six DOFs PMs. Using the parallelograms to realize the kinematic decoupled. Li et al. [22] proposed a R-Cube decoupled 3T PM. Briot et al. [23] synthesized a faster isotropic 3T PM. Awtar [24] proposed a geometrically decoupled XYZ flexible PM. Carricato [25] realized the type synthesis of a fully isotropic 3T1R PM. Briot et al. [26] designed a series of decoupled PMs using a 4U parallelogram mechanism. Jin et al. [27] utilized the parallelogram mechanism and its evolved structures to synthesize decoupled 2T2R PMs, the moving platforms of which have a large rotation angle. Sun et al. [28] designed a 2-DOF flexible fully decoupled PM driven by piezoelectric ceramics using 4-PP double bending bars as structural joints of prism. Gao et al. [29] synthesized decoupled 3T PMs using G_F set theory. Zhang et al. [30] performed kinematic and singularity analysis of decoupled 2T1R PMs. Qu et al. [31] synthesized decoupled 2T1R PMs based on screw theory and Lie group. Jin et al. [32] utilized single open chain structural feature blended with topology structure and scale parameters to design decoupled 3-DOF PMs. Shen et al. [33] proposed a systematic methodology for synthesizing kinematic structures of 3-DOF weakly-coupled translational parallel mechanisms based on hybrid chains.

It can be seen that the decoupled 3T PMs have been widely used in many fields, but they are difficult to be applied in the high stiffness situations [34]. Zhao et al. [35] introduced the closed-loop structure units into the kinematic chains of the mechanism, and took the closed-loop structure as the driving unit or the connecting unit in the limb, so that the stiffness performance of the mechanism could be improved because of the scale characteristics and closed characteristics of the closed-loop structure.

For improving the stiffness of the fully decoupled PMs, a new synthesis method of the fully decoupled 3T PMs with closed-loop units is proposed. The structural form of the closed-loop unit is geometrically closed. The unit is a closed loop formed by the basic decoupled limb, the constructed limb, the moving platform and the fixed platform. The criterion of full decoupled of the 3T PM is presented, in which the reciprocal product of the output twist screws and the transmission wrench screws is a diagonal matrix (the main diagonal elements are all nonzero constants). A synthetic process is developed with the closed-loop unit construction method, and a mass of high-stiffness fully decoupled 3T PMs with closed-loop units are synthesized. The synthesized mechanism not only has the 3T fully decoupled characteristic, but also has the high stiffness, and has a good application prospect in the field of PM with high stiffness.

2 Full Decoupled Condition of Three Translational Parallel Mechanism

According to the definition of screw, the composition form of the screw is (S; S₀), where S is called the original part and represents the direction vector, ${\varvec{S}}_{0}$ is called the dual part and can be expressed as:

$${\varvec{S}}_{0} = {\varvec{r}} \times {\varvec{S}} + h{\varvec{S}},$$

(1)

where r is the position vector from the origin of the coordinate to any point on $\boldsymbol{\$}$, and h is the pitch of the screw. The screw can also be expressed in Plücker coordinate $\left( L M N ;P Q R \right)$. The screw has two special forms: When the pitch h=0, the screw degenerates into a line vector $\boldsymbol{\$ } = \left[ {{\varvec{S}}{;}}\, {{\varvec{r}} \times {\varvec{S}}} \right]^{\text T}$, which can represent a motion screw of a revolute pair or a pure line force vector applied to a rigid body; when h tends to infinity, the screw is changed to a couple vector $\boldsymbol{\$ } = \left[ {{0;}} {\varvec{S}} \right]^{\text T}$, which can represent a motion screw of a prismatic pair or a pure couple force vector applied to a rigid body. If the two screws $\boldsymbol{\$ } _{1}\left( {{\varvec{S}}{;}} {{\varvec{S}}_{0} } \right)$ and $\boldsymbol{\$ } _{2}$$\left( {{\varvec{S}}^{r} {;}} {{\varvec{S}}_{0}^{r} } \right)$ meet the conditions:

$$\boldsymbol{\$ } _{{1}} \circ \boldsymbol{\$ }_{{2}} = {\varvec{S}} \cdot {\varvec{S}}_{0}^{r} + {\varvec{S}}^{r} \cdot {\varvec{S}}_{0} ,$$

(2)

then screw $\boldsymbol{\$ } _{1}$ and screw $\boldsymbol{\$ } _{2}$ are mutually anti-screws, where “o” represents the reciprocal product of the two screws, and “·” represents the dot multiplication of the two vectors.

Transmission wrench screw refers to the constraint wrench screw that has zero reciprocal product with all other twist screw except the actuated screw of the limb [36]. The transmission wrench screw transmits the motion/force of the input joint to the moving platform to actuate the platform in an expected motion.

Therefore, to achieve full decoupled of the three translational directions of the PM, the translational motion in one output DOF direction is only controlled by transmission wrench screw of one limb, and it will not be affected by the other two limbs. According to the screw theory, the physical meaning of the reciprocal product of the force screw and the twist screw is the instantaneous power generated by the force screw to the twist screw. In this paper, a reciprocal product algorithm of screw matrices is defined:

$${\varvec{A}} = \left[ {\begin{array}{*{20}c} {\boldsymbol{\$ }_{{T{1}}} } \\ {\boldsymbol{\$ }_{{T{2}}} } \\ \vdots \\ {\boldsymbol{\$ }_{Tn} } \\ \end{array} } \right]_{n \times 1} {\varvec{B}} = \left[ {\begin{array}{*{20}c} {\boldsymbol{\$ }_{{1}} } \\ \begin{gathered} \boldsymbol{\$ }_{{2}} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots \hfill \\ \boldsymbol{\$ }_{n} \hfill \\ \end{gathered} \\ \end{array} } \right]^{\text T}_{{\text{1} \times n}} ,$$

(3)

$${\varvec{A}} \circ {\varvec{B}} = {\varvec{C}} = \left[ {\begin{array}{*{20}c} {\boldsymbol{\$ }_{{T{1}}} \circ \boldsymbol{\$ }_{{1}} } {\boldsymbol{\$ }_{{T{1}}} \circ \boldsymbol{\$ }_{{2}} } \cdots {\boldsymbol{\$ }_{{T{1}}} \circ \boldsymbol{\$ }_{n} } \\ {\boldsymbol{\$ }_{{T\text{2}}} \circ \boldsymbol{\$ }_{{1}} } {\boldsymbol{\$ }_{{T\text{2}}} \circ \boldsymbol{\$ }_{{2}} } \cdots {\boldsymbol{\$ }_{{T\text{2}}} \circ \boldsymbol{\$ }_{n} } \\ \vdots \vdots \ddots \vdots \\ {\boldsymbol{\$ }_{Tn} \circ \boldsymbol{\$ }_{{1}} } {\boldsymbol{\$ }_{Tn} \circ \boldsymbol{\$ }_{{2}} } \cdots {\boldsymbol{\$ }_{Tn} \circ \boldsymbol{\$ }_{n} } \\ \end{array} } \right]_{n \times n} ,$$

(4)

where A is a matrix of transmission wrench screws, B is a matrix of output twist screws, and C is the reciprocal product matrix of A and B. The elements in row i and column j in matrix C can be expressed as:

$$\left( {{\varvec{A}} \circ {\varvec{B}}} \right)_{{{{ij}}}} = \boldsymbol{\$ }_{{{{Ti}}}} \circ \boldsymbol{\$ }_{{{j}}} ,$$

(5)

where $\boldsymbol{\$ }_{{{{Ti}}}}$ is the transmission wrench screw of the i limb, and $\boldsymbol{\$ }_{{{j}}}$ is the output twist screw of the j limb.

For a fully decoupled PM, there is only instantaneous power between the transmission wrench screw and the corresponding output twist screw. Based on this, the condition for full decoupled of the 3T PM is proposed: The instantaneous power matrix is obtained by reciprocal product of the transmission wrench screw matrix of the mechanism and the corresponding output twist screw matrix, and it is a diagonal matrix (none of the main diagonal elements is zero), as shown in Eq. (6):

$$\left[ {\begin{array}{*{20}c} {\boldsymbol{\$ }_{{{{T}}1}} } \\ {\boldsymbol{\$ }_{{{{T}}2}} } \\ {\boldsymbol{\$ }_{{{{T}}3}} } \\ \end{array} } \right] \circ \left[ {\begin{array}{*{20}c} {\boldsymbol{\$ }_{1} } \\ {\boldsymbol{\$ }_{2} } \\ {\boldsymbol{\$ }_{3} } \\ \end{array} } \right]^{\text T} = \left[ {\begin{array}{*{20}c} {\boldsymbol{\$ }_{{{{T}}1}} \circ \boldsymbol{\$ }_{1} } & 0 & 0 \\ 0 & {\boldsymbol{\$ }_{{{{T}}2}} \circ \boldsymbol{\$ }_{2} } & 0 \\ 0 & 0 & {\boldsymbol{\$ }_{{{{T}}3}} \circ \boldsymbol{\$ }_{3} } \\ \end{array} } \right]_{3 \times 3} .$$

(6)

3 Construction Method of Closed-Loop Units

In this paper, a closed-loop unit is constructed in the limb to increase the stiffness of the fully decoupled PMs, and it will not change the decoupled characteristic. According to the relationship between the output twist screw and transmission wrench screw in the fully decoupled condition, the transmission wrench screw of the limb is determined. Next, the basic decoupled limb is synthesized by combining the construction method of the actuated and unactuated screw. In this paper, the actuated and unactuated screw refer to the twist screw. Finally, the limb which is completely consistent and driven synchronously with the basic decoupled limb is generated, and it forms a closed-loop unit with the basic decoupled limb. The construction method of the closed-loop unit needs to consider the types of kinematic pairs which are directly connected to the moving platform and the fixed platform. There are two types of kinematic pairs, prismatic pair and revolute pair. The spatial geometric relationship of two kinematic pairs of the basic decoupled limb and the constructed limb are also considered. The spatial geometric relationship of kinematic pairs can be divided into three cases: coaxial, parallel and share. When the driving pairs are shared, the redundant actuation is not necessary to be added. When the driving pairs are coaxial or parallel, redundant actuation synchronized with the actuation of the basic decoupled limb needs to be added in the construction limb.

The construction method of the closed-loop units is as follows.

• 1. Using prismatic pair as the driving pair

  When the kinematic pair directly connected to the moving platform in the basic decoupled limb is revolute (R) pair, the R pair connected to the moving platform in the constructed limb can be coaxial or parallel to it. When the R pairs are coaxial, the driving pair in the constructed limb and the one in the basic decoupled limb can be coaxial (as shown in Figure 1a), or shared (as shown in Figure 1b). There is no need to add redundant actuation only when the driving pair is shared. When the driving pair in the constructed limb is coaxial to the one in the basic decoupled limb, the redundant actuation should be added into the closed-loop unit, and it is synchronized with the actuation of the basic decoupled limb. When the R pairs are parallel, the driving pair in the constructed limb is parallel to the one in the basic decoupled limb. The redundant actuation should be added into the closed-loop unit, and it is synchronized with the actuation of the basic decoupled limb, as shown in Figure 1c.

  When the kinematic pair directly connected to the moving platform in the basic decoupled limb is prismatic (P) pair, the P pair connected to the moving platform in the constructed limb is coaxial or parallel to itself. They can also share the same P pair.

  When the P pair is shared, the driving pair in the constructed limb and the one in the basic decoupled limb can be shared (as shown in Figure 1d), coaxial (as shown in Figure 1e) or parallel (as shown in Figure 1f). There is no need to add redundant actuation only when the driving pair is shared.

  When the P pairs are parallel, the driving pair in the constructed limb and the one in the basic decoupled limb can be shared (as shown in Figure 1g), or coaxial (as shown in Figure 1h). When the driving pair in the constructed limb is coaxial to the one in the basic decoupled limb, the redundant actuation should be added into the closed-loop unit, and it is synchronized with the actuation of the basic decoupled limb.

  When the P pairs are coaxial, the driving pair in the constructed limb is parallel to the one in the basic decoupled limb. In the above case, the redundant actuation should be added into the closed-loop unit, and it is synchronized with the actuation of the basic decoupled limb, as shown in Figure 1i.

• 2. Using revolute pair as the driving pair

  When the kinematic pair directly connected to the moving platform in the basic decoupled limb is R pair, the R pair connected to the moving platform in the constructed limb can only be parallel to it. When two R pairs are parallel, the driving pair in the constructed limb and the one in the basic decoupled limb can be coaxial (as shown in Figure 2a), parallel (as shown in Figure 2b) or shared (as shown in Figure 2c). There is no need to add redundant actuation only when the driving pair is shared.

  When the kinematic pair directly connected to the moving platform in the basic decoupled limb is P pair, the P pair connected to the moving platform in the constructed limb is coaxial (as shown in Figure 2e), parallel (as shown in Figure 2f) or shared with it (as shown in Figure 2d). When the P pair is shared or coaxial, the driving pair in the constructed limb is parallel to the one in the basic decoupled limb. When the P pairs are parallel, the driving pair in the constructed limb is coaxial to the one in the basic decoupled limb. In the above case, the redundant actuation should be added into the closed-loop unit, and it is synchronized with the actuation of the basic decoupled limb.

Figure 1

Using prismatic pair as the driving pair

Figure 2

Using revolute pair as the driving pair

4 A New Synthetic Method of Three Translational Fully Decoupled Parallel Mechanism with Closed-Loop Units

From the above analysis, the synthesis process of the fully decoupled 3T PMs is developed as follows, and the flow chart is shown in Figure 3.

• 1. Based on the DOF and decoupled requirements of the 3T PM, a full decoupled condition is proposed that the reciprocal product of the transmission wrench screw matrix and the output twist screw matrix of the limb is a diagonal matrix (none of the main diagonal elements are zero), and the forms of transmission wrench screw of each limb are determined.

• 2. The form of the actuated screw in the limb is obtained according to the criterion that the reciprocal product of the transmission wrench screw and the actuated screw in the same limb is a nonzero constant.

• 3. The form of the unactuated screw is determined according to the criterion that the reciprocal product of the transmission wrench screw and all the other kinematic screws except the actuated screw in the same limb is equal to zero.

• 4. On the basis of the type of actuated screw and unactuated screw, the connectivity of the limb and the different order of the kinematic pairs, the basic decoupled limb is synthesized.

• 5. The limb which is completely consistent and driven synchronously with the basic decoupled limb is constructed by the closed-loop unit construction method. Then, a closed-loop unit with a basic decoupled limb is constituted to improve the stiffness performance of the mechanism, and the type synthesis of the fully decoupled 3T PM with high stiffness is realized.

Figure 3

Synthesis flow chart of three translational fully decoupled parallel mechanism

5 Synthesis of Three Translational Fully Decoupled Parallel Mechanism with Closed-Loop Units

Decoupled limbs need to be constructed firstly according to the expected DOF and decoupled characteristic for synthesis of the high-stiffness decoupled 3T PMs with the closed-loop units. Under the condition that the decoupled characteristic of the mechanism is not changed, a closed-loop unit is formed with the basic decoupled limb and the constructed limb based on the closed-loop unit construction method. Ultimately, a high-stiffness decoupled 3T PM with the closed-loop units is constituted.

5.1 Configuration Synthesis of Limb 1

As shown in Figure 4, o-xyz is the base coordinate system, $\boldsymbol{\$ }_{1} ,\boldsymbol{\$ }_{2} ,\boldsymbol{\$ }_{3}$ are the output twist screws of the three motion directions of the moving platform. $\boldsymbol{\$ }_{{{{Ti}}}}$ is the transmission wrench screw of the limb i acting on the moving platform. According to the criterion for the full decoupled of PMs, each limb of the fully decoupled 3T PM only controls one output translation motion. Assuming that the three limbs control the output motion in the three directions of x, y and z axes respectively, and the output twist screw is a translational screw with infinite pitch in the same direction as the output motion. Therefore, $\boldsymbol{\$ }_{1} ,\boldsymbol{\$ }_{2} ,\boldsymbol{\$ }_{3}$ are expressed as:

$$\begin{gathered} \boldsymbol{\$ }_{1} = \left( {\begin{array}{*{20}c} 0 0 0 \\ \end{array} ;\begin{array}{*{20}c} 1 0 0 \\ \end{array} } \right), \hfill \\ \boldsymbol{\$ }_{2} = \left( {\begin{array}{*{20}c} 0 0 0 \\ \end{array} ;\begin{array}{*{20}c} 0 1 0 \\ \end{array} } \right), \hfill \\ \boldsymbol{\$ }_{3} = \left( {\begin{array}{*{20}c} 0 0 0 \\ \end{array} ;\begin{array}{*{20}c} 0 0 1 \\ \end{array} } \right). \hfill \\ \end{gathered}$$

(7)

Figure 4

Schematic diagram of transmission screw and actuated screw of limb 1

Based on the decoupled criterion, the output translational motion along the x-axis direction is only provided by limb 1, that is, the reciprocal product of the transmission wrench screw of the limb and the corresponding output twist screw is not zero, and the reciprocal product with the output twist screw controlled by other limbs is equal to zero.

Provided that the transmission wrench screw of limb 1 is

$$\boldsymbol{\$ }_{{{{T}}1}} = \left( {{{a}}_{1} } {{{b}}_{1} } {{{c}}_{1} ;\begin{array}{*{20}c} {{{d}}_{1} } {{{e}}_{1} } {{{f}}_{1} } \end{array} } \right),$$

(8)

If $\boldsymbol{\$ }_{{{{T}}1}} \circ \boldsymbol{\$ }_{1} \ne 0,\boldsymbol{\$ }_{{{{T}}1}} \circ \boldsymbol{\$ }_{2} = 0,\boldsymbol{\$ }_{{{{T}}1}} \circ \boldsymbol{\$ }_{3} = 0$ are satisfied, the following equations can be deduced:

$$\begin{gathered} {{a}}_{1} + {{d}}_{1} \cdot 0 + {{e}}_{1} \cdot 0 + {{f}}_{1} \cdot 0 \ne 0, \hfill \\ {{b}}_{1} + {{d}}_{1} \cdot 0 + {{e}}_{1} \cdot 0 + {{f}}_{1} \cdot 0 = 0, \hfill \\ {{c}}_{1} + {{d}}_{1} \cdot 0 + {{e}}_{1} \cdot 0 + {{f}}_{1} \cdot 0 = 0. \hfill \\ \end{gathered}$$

(9)

It can be solved from Eq. (9) that ${{b}}_{1} ,{{c}}_{1}$ must be zero and ${{a}}_{1} \ne 0$. Eq. (9) will hold true no matter what the values of $\boldsymbol{\$ }_{{{{Ti}}}}$ are. The value of ${{d}}_{1} = 0,{{e}}_{1} \ne 0,{{f}}_{1} \ne 0$ is taken. The transmission wrench screw does not pass through the origin of the coordinate. In this case, the transmission wrench screw acting on the moving platform of limb 1 can be finally determined as:

$$\boldsymbol{\$ }_{{{{T}}1}} = \left( 1\; 0\; 0 ; 0 \; {{{e}}_{1} }\; {{{f}}_{1} } \right).$$

(10)

It can be known from above analysis that the relationship between transmission wrench screw of each limb and output twist screw of the moving platform of the fully decoupled 3T PM is that the transmission wrench screw of each limb is coplanar with the output twist screw of the corresponding limb, and perpendicular to the output twist screw of other limbs.

The transmission wrench screw $\boldsymbol{\$ }_{{{{T}}1}}$ of limb 1 is the force screw in the direction of the x-axis, which is in the same direction as the output motion controlled by the limb. Similarly, limb 2 controls the output motion in the y-axis direction, and limb 3 controls the output motion in the z-axis direction. Thus, the transmission wrench screw of limb 2 is the force screw in the y-axis direction, and the transmission wrench screw of limb 3 is the force screw in the z-axis direction. The transmission wrench screw of the limb acting on the moving platform can only be the linear force vector in the same direction as the output twist screw of the limb.

According to the condition that the reciprocal product of the transmission wrench screw $\boldsymbol{\$ }_{{{{Ti}}}}$ and the actuated screw $\boldsymbol{\$ }_{{{{i}}1}}$ in the same kinematic limb is a nonzero constant: $\boldsymbol{\$ }_{{{{Ti}}}} \circ \boldsymbol{\$ }_{{{{i}}1}} \ne 0$ (i is the i limb), the actuated screw of each limb is constructed. Take limb 1 as an example.

Provided that the actuated screw $\boldsymbol{\$ }_{11}$ of limb 1 is

$$\boldsymbol{\$ }_{{\text{11}}} \text{ = }\left( {\begin{array}{*{20}c} {{{l}}_{\text{1}} } \; {{{m}}_{\text{1}} } \;{{{n}}_{\text{1}} \text{;}\begin{array}{*{20}c} {{{u}}_{\text{1}} }\; {{{v}}_{\text{1}} } \;{{{w}}_{\text{1}} } \\ \end{array} } \\ \end{array} } \right).$$

(11)

If $\boldsymbol{\$ }_{{{{T}}1}} \circ \boldsymbol{\$ }_{11} \ne 0$, the following equation needs to be satisfied:

$${{l}}_{1} \cdot 0 + {{m}}_{1} e_{1} + {{n}}_{1} {{f}}_{1} + {{u}}_{1} + {{v}}_{1} \cdot 0 + {{w}}_{1} \cdot 0 \ne 0.$$

(12)

Eq. (12) will hold true no matter what the values of ${{l}}_{1} ,{{v}}_{1} ,{{w}}_{1}$ are.

If the pitch of $\boldsymbol{\$ }_{11}$ is infinitely large, there's only one case, which is ${{m}}_{1} = 0,{{n}}_{1} = 0,{{u}}_{1} \ne 0$.

$$\boldsymbol{\$ }_{11}^{1} = \left( {\begin{array}{*{20}c} 0\; 0\; 0 \\ \end{array} ;\begin{array}{*{20}c} 1 \; 0\; 0 \\ \end{array} } \right).$$

(13)

At this moment, the driving pair is a P pair with the axis along the x-axis direction.

If the pitch of $\boldsymbol{\$ }_{11}$ is zero, there are two cases:

A. ${{m}}_{1} \ne 0,{{n}}_{1} = 0,{{u}}_{1} \ne 0$

$$\boldsymbol{\$ }_{11}^{2} = \left( {\begin{array}{*{20}c} 0\, 1\, 0 \\ \end{array} ;\begin{array}{*{20}c} {{{u}}_{1} } \, 0 \,{{{w}}_{1} } \\ \end{array} } \right).$$

(14)

At this moment, the driving pair is a R pair with the axis along the y-axis direction.

B. ${{m}}_{1} = 0,{{n}}_{1} \ne 0,{{u}}_{1} \ne 0$

$$\boldsymbol{\$ }_{11}^{3} = \left( {\begin{array}{*{20}c} 0\, 0\, 1 \\ \end{array} ;\begin{array}{*{20}c} {{{u}}_{1} }\, {{{v}}_{1} }\, 0 \\ \end{array} } \right).$$

(15)

At this moment, the driving pair is a R pair with the axis along the z-axis direction.

To sum up, the transmission wrench screw $\boldsymbol{\$ }_{{{{T}}{1}}}$ of limb 1 acting on the moving platform and all possible actuated screws $\boldsymbol{\$ }_{11}^{1}$, $\boldsymbol{\$ }_{11}^{2}$, $\boldsymbol{\$ }_{11}^{3}$ are shown in Figure 4.

According to the condition that the reciprocal product of the transmission wrench screw $\boldsymbol{\$ }_{{{{Ti}}}}$ and the unactuated kinematic screw except the actuated screw in the same limb is equal to zero: $\boldsymbol{\$ }_{{{{Ti}}}} \circ \boldsymbol{\$ }_{{{{ij}}}} = 0$ (i represents the i limb, j represents other kinematic screws in the limb except the actuated screw and $1 < {{j}} \le 5$), the unactuated screw of limb 1 is

$$\boldsymbol{\$ }_{{1{{j}}}} = ({{{l}}_{{1{{j}}}} }\; {{{m}}_{{1{{j}}}} } \;{{{n}}_{{1{{j}}}} } ; {{{u}}_{{1{{j}}}} }\; {{{v}}_{{1{{j}}}} } \;{{{w}}_{{1{{j}}}} } ).$$

(16)

If $\boldsymbol{\$ }_{{{{T}}1}} \circ \boldsymbol{\$ }_{{1{{j}}}} = 0$, the following equation can be deduced:

$${{l}}_{{1{{j}}}} \cdot 0 + {{m}}_{{1{{j}}}} {{e}}_{1} + {{n}}_{{1{{j}}}} {{f}}_{1} + {{u}}_{{1{{j}}}} + {{v}}_{{1{{j}}}} \cdot 0 + {{w}}_{{1{{j}}}} \cdot 0 = 0.$$

(17)

There are two cases for Eq. (17) to be true. In the first case, the conditions ${{m}}_{{1{{j}}}} = 0,{{n}}_{{1{{j}}}} = 0,{{u}}_{{1{{j}}}} = 0$ need to be satisfied, the unactuated screw in the limb may be a line vector in the x-axis direction or a couple vector in the direction of y and z axis. In the second case, $\boldsymbol{\$ }_{{1{{j}}}}$ is a line vector which always intersects with $\boldsymbol{\$ }_{{{{T}}1}}$, and both cases can coexist in the same limb.

Since the limbs should contain all the motion characteristics of the moving platform to achieve the expected output motion of the moving platform, each limb must have translational DOF in at least x, y and z directions, and the connectivity of the limb should be at least 3. When the actuation is a R pair, 2R parallel sub-chain can be used as the driving pair, but the original limb should contain translational DOF whose direction is perpendicular to the axis of the R pair (a 2R parallel sub-chain can be considered as a linear combination of a R pair and a P pair). Moreover, there should be one translational DOF parallel to the negative motion direction of the 2R sub-chain. Negative motion direction is the non-corresponding output DOF direction of the two translational DOFs vertical to the axis of R pair. As shown in Table 1, the types of unactuated screw of limb 1 and their satisfying conditions are discussed according to the types of driving pairs of limb 1.

Table 1 Unactuated screw types in limb 1 and their satisfying conditions

In this paper, only the connectivity of 3, 4 and 5 of limb 1 are considered. According to the types of actuated and unactuated screws, the configuration orientation and connectivity of limb 1, all possible kinematic limb structures are synthesized, as shown in Table 2, P represents the prismatic pair, R represents the revolute pair, x, y, z represent the axis directions of the prismatic pair or revolute pair, C represents the cylindrical pair, U represents the universal pair, and the lower angle symbol n indicates that the direction of the prismatic pair is orthogonal to the directions of the adjacent two R pairs. The 2R parallel sub-chain can be replaced by a parallelogram structure Pa, which is composed of four R pairs in the same direction as the actuated screw to improve bearing capacity of the limb. To simplify the structure, suppose that the axes of the adjacent kinematic pairs in the limb are parallel or perpendicular to each other. This configuration can be designed and reformed according to the specific application conditions in the actual engineering. Schematic diagrams of typical limb configurations and structures are shown in Table 3. The basic decoupled limbs in Tables 2 and 3 are not the final configurations. The 3T fully decoupled PMs with closed-loop units are formed by the redundant limbs constructed by the new method of closed-loop unit construction and the basic decoupled limbs.

Table 2 Configuration of limb 1

Table 3 Typical configuration and structure diagram of limb 1

5.2 Configuration Synthesis of Limb 2

If limb 2 provides y-direction output motion, provided that the transmission wrench screw of limb 2 is

$$\boldsymbol{\$ }_{{{{T}}2}} = \left( {\begin{array}{*{20}c} {{{a}}_{2} }\, {{{b}}_{2} }\, {{{c}}_{2} ;\begin{array}{*{20}c} {{{d}}_{2} }\, {{{e}}_{2} }\, {{{f}}_{2} } \\ \end{array} } \\ \end{array} } \right),$$

(18)

if condition $\boldsymbol{\$ }_{{{{T}}2}} \circ \boldsymbol{\$ }_{2} \ne 0,\boldsymbol{\$ }_{{{{T}}2}} \circ \boldsymbol{\$ }_{1} = 0,\boldsymbol{\$ }_{{{{T}}2}} \circ \boldsymbol{\$ }_{3} = 0$ is satisfied, the following equations can be deduced:

$$\begin{gathered} {{b}}_{2} + {{d}}_{2} \cdot 0 + {{e}}_{2} \cdot 0 + {{f}}_{2} \cdot 0 \ne 0, \hfill \\ {{a}}_{2} + {{d}}_{2} \cdot 0 + {{e}}_{2} \cdot 0 + {{f}}_{2} \cdot 0 = 0, \hfill \\ {{c}}_{2} + {{d}}_{2} \cdot 0 + {{e}}_{2} \cdot 0 + {{f}}_{2} \cdot 0 = 0. \hfill \\ \end{gathered}$$

(19)

It can be solved from Eq. (19) that ${{a}}_{2} ,{{c}}_{2}$ must be zero and ${{b}}_{2} \ne 0$. Eq. (19) will hold true no matter what the values of ${{d}}_{2} ,{{e}}_{2} ,{{f}}_{2}$ are. The value of ${{e}}_{2} = 0,{{d}}_{2} \ne 0,{{f}}_{2} \ne 0$ is taken. In this case, the transmission wrench screw acting on the moving platform of limb 2 can be finally determined as:

$$\boldsymbol{\$ }_{{{{T}}\text{2}}} = \left( {\begin{array}{*{20}c} 0\, 1\, 0 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{2} }\, 0\, {{{f}}_{2} } \\ \end{array} } \right).$$

(20)

The transmission wrench screw ${{\$ }}_{{{{T}}{2}}}$ of limb 2 is the force screw in the y-axis direction, which is in the same direction as the output motion of the limb.

According to the conditions that the reciprocal product of the transmission wrench screw $\boldsymbol{\$ }_{{{{Ti}}}}$ and the actuated screw $\boldsymbol{\$ }_{{{{i}}1}}$ in the same kinematic limb is a nonzero constant: $\boldsymbol{\$ }_{{{{Ti}}}} \circ \boldsymbol{\$ }_{{{{i}}1}} \ne 0$ (i is the i limb), the actuated screw of each limb is obtained.

Provided that the actuated screw $\boldsymbol{\$ }_{21}$ of limb 2 is

$$\boldsymbol{\$ }_{{\text{21}}} { = }\left( {\begin{array}{*{20}c} {{{l}}_{2} }\; {{{m}}_{2} }\; {{{n}}_{2} {;}\begin{array}{*{20}c} {{{u}}_{2} }\; {{{v}}_{2} } \;{{{w}}_{2} } \\ \end{array} } \\ \end{array} } \right),$$

(21)

if $\boldsymbol{\$ }_{{{{T2}}}} \circ \boldsymbol{\$ }_{21} \ne 0$, the following equation needs to be satisfied:

$${{l}}_{2} \cdot {{d}}_{2} + {{m}}_{2} \cdot 0 + {{n}}_{2} {{f}}_{2} + {{u}}_{2} \cdot 0 + {{v}}_{2} + {{w}}_{2} \cdot 0 \ne 0.$$

(22)

Eq. (22) will hold true no matter what the values of ${{m}}_{2} ,{{u}}_{2} ,{{w}}_{2}$ are.

If the pitch of $\boldsymbol{\$ }_{21}$ is infinitely large, there's only one case, ${{l}}_{2} = 0,{{n}}_{2} = 0,{{v}}_{2} \ne 0,$

$$\boldsymbol{\$ }_{21}^{1} = \left( {\begin{array}{*{20}c} 0\, 0 \, 0 \\ \end{array} ;\begin{array}{*{20}c} 0\, 1\, 0 \\ \end{array} } \right).$$

(23)

At this moment, the driving pair is a P pair with the axis along the y-axis direction.

If the pitch of $\boldsymbol{\$ }_{21}$ is zero, there are two cases, which are:

A. ${{l}}_{2} \ne 0,{{n}}_{2} = 0,{{v}}_{2} \ne 0$

$$\boldsymbol{\$ }_{21}^{2} = \left( 1\, 0\, 0 ;0\; {{{v}}_{2} } \,{{{w}}_{2} } \right).$$

(24)

At this moment, the driving pair is a R pair with the axis along the x-axis direction.

B. ${{l}}_{2} = 0,{{n}}_{2} \ne 0,{{v}}_{2} \ne 0$

$$\boldsymbol{\$ }_{21}^{3} = \left( 0\, 0\, 1 ; {{{u}}_{2} } \,{{{v}}_{2} }\, 0 \right).$$

(25)

At this moment, the driving pair is a R pair with the axis along the z-axis direction. To sum up, the transmission wrench screw $\boldsymbol{\$ }_{{{{T}}2}}$ of limb 2 acting on the moving platform and all possible actuated screws $\boldsymbol{\$ }_{21}^{1}$, $\boldsymbol{\$ }_{21}^{2}$, $\boldsymbol{\$ }_{21}^{3}$ are shown in Figure 5.

Figure 5

Schematic diagram of transmission force screw and actuated screw of limb 2

On the basis of the condition that the reciprocal product of the transmission wrench screw $\boldsymbol{\$ }_{{{{Ti}}}}$ and the unactuated kinematic screw except the actuated screw in the same limb is equal to zero: $\boldsymbol{\$ }_{{{{Ti}}}} \circ \boldsymbol{\$ }_{{{{ij}}}} = 0$(i represents the i limb, j represents other kinematic screws in the limb except the actuated screw and $1 < {{j}} \le 5$), the unactuated screw of limb 2 is

$$\boldsymbol{\$ }_{{2{{j}}}} = (\begin{array}{*{20}c} {{{l}}_{{2{{j}}}} } \;{{{m}}_{{2{{j}}}} }\; {{{n}}_{{2{{j}}}} } \\ \end{array} ;\begin{array}{*{20}c} {{{u}}_{{2{{j}}}} } \;{{{v}}_{{2{{j}}}} } \;{{{w}}_{{2{{j}}}} } \\ \end{array} ).$$

(26)

If $\boldsymbol{\$ }_{{{{T}}2}} \circ \boldsymbol{\$ }_{{2{{j}}}} = 0$, the following equation can be deduced:

$${{l}}_{{2{{j}}}} d_{2} + {{m}}_{{2{{j}}}} \cdot 0 + {{n}}_{{2{{j}}}} {{f}}_{2} + {{u}}_{{2{{j}}}} \cdot 0 + {{v}}_{{2{{j}}}} + {{w}}_{{2{{j}}}} \cdot 0 = 0.$$

(27)

There are two cases for Eq. (27) to be true. In the first case, the condition ${{l}}_{{2{{j}}}} = 0,{{n}}_{{2{{j}}}} = 0,{{v}}_{{2{{j}}}} = 0$ needs to be satisfied. The unactuated screw can be a line vector in the y-axis direction or a couple vector in the direction of x and z axis. In the second case, $\boldsymbol{\$ }_{{2{{j}}}}$ is a line vector which always intersects with $\boldsymbol{\$ }_{{{{T}}2}}$, and both cases can coexist in the same limb.

The types and conditions of the unactuated kinematic screw in limb 2 are similar to those in Table 1. In Table 1, the actuated screw $\boldsymbol{\$ }_{11}^{1}$ is replaced by $\boldsymbol{\$ }_{21}^{1}$, transmission wrench screw $\boldsymbol{\$ }_{{{{T}}1}}$ is replaced by $\boldsymbol{\$ }_{{{{T}}2}}$. For axis directions of all kinematic pairs, x is replaced by y, y is replaced by z, and z is replaced by x.

The construction method of limb 2 is the same as that of limb 1, just replace all the subscripts in Table 2 as follows: x is replaced with y, y is replaced with z, z is replaced with x, and all configurations of limb 2 can be obtained.

5.3 Configuration Synthesis of Limb 3

Assuming that the output motion along the z direction of the moving platform is only provided by the limb 3, the reciprocal product of the transmission wrench screw acting on the moving platform and the output twist screw in the z-axis direction is not zero, and the reciprocal product of the output twist screw in the other two directions is equal to zero. The derivation method of transmission wrench screw of limb 3 and the construction method of the limb are the same as those of limb 1, just replace the subscript of all the kinematic pairs in Table 2 with the following: x is replaced with z, y is replaced with x, z is replaced with y.

5.4 Construction Example of Closed-Loop Units

According to the closed-loop unit construction method proposed above, this paper select four kinds of basic decoupled limb configuration with compact structure and excellent performance in Table 2. 10 typical fully decoupled 3T PMs with high stiffness are constructed by connecting the closed-loop units containing above four kinds of basic decoupled limbs with the moving and fixed platforms, as shown in Table 4.

Table 4 Typical fully decoupled 3T PMs

The specific construction method of the closed-loop unit including the above four kinds of basic decoupled limbs is shown in Table 5.

Table 5 Criterion for construction of closed-loop unit for three translational fully decoupled parallel mechanisms

6 Synthetic Example and Verification of a Fully Decoupling Three Translational Parallel Mechanism with High Stiffness

The basic decoupled limb of the PRPR with a connectivity of 4 is selected as the optimal limb, and the P pair connected to the fixed platform is selected as the driving pair. Three closed-loop units are constructed and connected with the moving and fixed platforms to obtain the fully decoupled 3-P2(RPR) PM with high stiffness, as shown in Figure 6.

Figure 6

3-P2(RPR) fully decoupled parallel mechanism with closed-loop unit

6.1 Verification of DOF

As shown in Figure 6, the screw system of closed-loop unit 1 is

$$\begin{gathered} \boldsymbol{\$ }_{11} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{12} = (\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & {{{e}}_{1} } & {{{f}}_{1} } \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{13} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & {{{e}}_{2} } & {{{f}}_{2} } \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{14} = (\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & {{{e}}_{3} } & {{{f}}_{3} } \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{15} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{16} = (\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & {{\text{e}}_{1} } & {{\text{f}}_{1} } \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{17} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & {{{e}}_{2} } & {{{f}}_{2} } \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{18} = (\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & {{{e}}_{3} } & {{{f}}_{3} } \\ \end{array} ). \hfill \\ \end{gathered}$$

(28)

The reciprocal screw system of closed-loop unit 1 is

$$\begin{gathered} \boldsymbol{\$ }_{11}^{{{r}}} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & 1 & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{12}^{{{r}}} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} ). \hfill \\ \end{gathered}$$

(29)

The screw system of closed-loop unit 2 is

$$\begin{gathered} \boldsymbol{\$ }_{21} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 \,&\, \,1 & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{22} = (\begin{array}{*{20}c} 0 & 1 & 0 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{4} } & 0 & {{{f}}_{4} } \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{23} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{5} } & 0 & {{{f}}_{5} } \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{24} = (\begin{array}{*{20}c} 0 & 1 & 0 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{6} } & 0 & {{{f}}_{6} } \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{25} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & 1 & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{26} = (\begin{array}{*{20}c} 0 & 1 & 0 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{4} } & 0 & {{{f}}_{4} } \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{27} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{5} } & 0 & {{{f}}_{5} } \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{28} = (\begin{array}{*{20}c} 0 & 1 & 0 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{6} } & 0 & {{{f}}_{6} } \\ \end{array} ). \hfill \\ \end{gathered}$$

(30)

The reciprocal screw system of closed-loop unit 2 is

$$\begin{gathered} \boldsymbol{\$ }_{21}^{{{r}}} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{22}^{{{r}}} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} ). \hfill \\ \end{gathered}$$

(31)

The screw system of closed-loop unit 3 is

$$\begin{gathered} \boldsymbol{\$ }_{31} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 \,\,\,& 0 \,\,\,& 1 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{32} = (\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{7} } & {{{e}}_{7} } & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{33} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{8} } & {{{e}}_{8} } & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{34} = (\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{9} } & {{{e}}_{9} } & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{35} = (\begin{array}{*{20}c} 0\, & \,0 &\, 0 \\ \end{array} ;\begin{array}{*{20}c} 0\, &\, 0 &\, 1 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{36} = (\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{7} } & {{\text{e}}_{7} } & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{37} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{8} } & {{{e}}_{8} } & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{38} = (\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} ;\begin{array}{*{20}c} {{{d}}_{9} } & {{{e}}_{9} } & 0 \\ \end{array} ). \hfill \\ \end{gathered}$$

(32)

The reciprocal screw system of closed-loop unit 3 is

$$\begin{gathered} \boldsymbol{\$ }_{31}^{{{r}}} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} ), \hfill \\ \boldsymbol{\$ }_{32}^{{{r}}} = (\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} ;\begin{array}{*{20}c} 0 & 1 & 0 \\ \end{array} ). \hfill \\ \end{gathered}$$

(33)

The DOF of the 3-P2(RPR) mechanism is calculated by the modified Kutzbach-Grubler formula:

$${{M}} = \text{6}({{n}} - {{g}} - 1) + \sum\limits_{i = 1}^{g} {{{f}}_{{{i}}} } + {{v = }}\text{3,}$$

(34)

where M is the DOF of the mechanism, n is the total number of components containing the racks, g is the number of kinematic pairs, f_i is the DOF of the ith kinematic pair, and v is the number of redundant constraints for a multi-loop PM without considering the common constraints.

The above reciprocal screws of the closed-loop units show that the constraint screws of the three closed-loop units restrict the rotational DOF of the moving platform in three directions, thus verifying that 3-P2(RPR) has three translational DOFs.

6.2 Validation of Jacobian Matrix

Assuming that the input distance of each driving pair on the base is l₁, l₂, l₃, and the corresponding output distance of the moving platform is x, y, z respectively. According to the structural characteristics of the 3-P2(RPR) decoupled PM, the relationship between input and output can be expressed as:

$$\begin{gathered} {{x}} = {{l}}_{1} , \hfill \\ {{y}} = {{l}}_{2} , \hfill \\ {{z}} = {{l}}_{3} . \hfill \\ \end{gathered}$$

(35)

The time-derivatives of the above equations are given by

$$\begin{gathered} {\dot{{x}}} = {\dot{{l}}}_{1} , \hfill \\ {\dot{{y}}} = {\dot{{l}}}_{2} , \hfill \\ {\dot{{z}}} = {\dot{{l}}}_{3} . \hfill \\ \end{gathered}$$

(36)

Eq. (36) can be written in the matrix form:

$$\left[ {\begin{array}{*{20}c} {{\dot{{x}}}} \\ {{\dot{{y}}}} \\ {{\dot{{z}}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\dot{{l}}}_{1} } \\ {{\dot{{l}}}_{2} } \\ {{\dot{{l}}}_{3} } \\ \end{array} } \right].$$

(37)

The Jacobian matrix of the mechanism can be obtained:

$${{J}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right].$$

(38)

The Jacobian matrix is a unit matrix, and the Jacobian determinant is always a fixed value. There is no singular position in the working space of the moving platform, and there will be no movement distortion. One output motion is controlled by only one corresponding input motion, which also verifies that the 3-P2(RPR) PM has three translational full decoupled characteristics, and the mechanism is completely isotropic in the entire working space. Under the condition of satisfying the decoupled characteristics, the isotropic PM has excellent kinematic characteristics and mechanical properties at the same time.

6.3 Stiffness Verification of a Three Translational Fully Decoupled Parallel Mechanism with Closed-Loop Units

The spatial composite deformation (bending, stretching and torsion deformation) of the connecting rod under the action of the generalized restraining force (including the driving force) is obtained. Firstly, the limb generalized constraint force stiffness matrix is solved. Then, combining with the static balance equation of the moving platform and the principle of virtual work, the overall stiffness matrix of the mechanism is obtained.

The analytical expression of 3-PRPR PM stiffness matrix is

$${\varvec{K}_{3 - \text{PRPR}}} = {\varvec{J}_1}{\varvec{K}_1}\varvec{J}_1^{\text T} + {\varvec{J}_2}{\varvec{K}_2}\varvec{J}_2^{\text T} + {\varvec{J}_3}{\varvec{K}_3}\varvec{J}_3^{\text T},$$

(39)

where ${\varvec{J}}_{{\text{i}}}$ represents the Jacobian matrix of the generalized constraint force of the $i$th limb, and ${\varvec{K}}_{{{i}}}$ represents the generalized constraint force stiffness matrix of the $i$th limb.

The analytical expression of the stiffness matrix of 3-P2(RPR) PM is

$$\begin{gathered} {{\varvec{K}}_{3 - \text{P2(RPR)}}} = {{\varvec{J}}_1}{{\varvec{K}}_1}{\varvec{J}}_1^{\text T} + {{\varvec{J}}_2}{{\varvec{K}}_2}{\varvec{J}}_2^{\text T} + {{\varvec{J}}_3}{{\varvec{K}}_3}{\varvec{J}}_3^{\text T} \hfill \\ + {{{\varvec{J}^\prime}}_1}{{{\varvec{K}^\prime}}_1}{\varvec{J}^\prime}_1^{\text T} + {{{\varvec{J}^\prime}}_2}{{{\varvec{K}^\prime}}_2}{\varvec{J}^\prime}_2^{\text T} + {{{\text{J}^\prime}}_3}{{{\varvec{K}^\prime}}_3}{\varvec{J}^\prime}_3^{\text T}, \hfill \\ \end{gathered}$$

(40)

where ${\varvec{J}^{\prime}}_{i}$ represents the Jacobian matrix of the generalized constraint force of the constructed limb in the $i$th loop. The direction of the constraint force of constructed limb and the basic decoupled limb are the same, so it can be deduced that ${\varvec{J}}_{i} = \varvec{J}^\prime_{i}$. The length and cross-sectional dimensions of the two RPR limbs in the loop are the same, so ${\varvec{K}}_{i} = \varvec{K}^\prime_{i}$ can be deduced. Therefore,

$${\varvec{K}}_{{3 - \text{P}2(\text{RPR)}}} = 2{\varvec{K}}_{{3 - \text{PRPR}}} .$$

(41)

The displacement of each driving pair is taken as ${q}_\text{i} = 0.05 - 0.4 \, \text{m}$, and the length of each rod of the RPR limb is ${{l}}_{{{i}}} = 0.5 - 0.75 \, \text{m}$. The outer diameter of the telescopic rod is ${{d}}_{1} = 0.045 \, \text{m}$, the inner diameter of the telescopic rod is ${{d}}_{2} = 0.030 \, \text{m}$. For comparison, the same constraint conditions of 3-PRPR and 3-P2(RPR) are given, and the same cross sections are selected in the z-axis direction (${{z}} = 0.45 \, \text{m}$) in the working space. The distribution of the maximum eigenvalues of the static stiffness matrix of 3-PRPR PM and 3-P2(RPR) PM is shown in Figure 7.

Figure 7

Distribution of the maximum eigenvalues of the static stiffness matrix of 3-PRPR PM and mechanism 3-P2 (RPR) PM

According to Figure 7, maximum eigenvalue of the stiffness matrix of the new 3-P2(RPR) PM is double larger than that of the 3-PRPR decoupled PM in the workspace. The addition of the closed-loop units doubles the stiffness of the PM, which verifies the 3-P2(RPR) PM has higher stiffness.

When the prismatic pair of 3-PRPR close to the moving platform is used as the driving pair, the mechanism is a coupling mechanism. After the above derivation, it can be seen that the number of constructed limbs (n) in the 3-Pn(RPR) loop is proportional to the stiffness of the mechanism. In order to analyze the stiffness difference between the 3-PRPR coupling mechanism [37] and the 3-Pn(RPR) decoupled mechanism, the VM index is used as the stiffness performance evaluation index, as shown in Eq. (42). That is, the reciprocal of the virtual work done by the external force. This performance index relates the value of the stiffness index to the external force and presents the ability of the mechanism to resist the external force in a specific direction.

$$\text{VM}=\frac{1}{\varvec{W}^\text{T},\delta \xi}=\frac{1}{\varvec{W}^\text{T}\varvec{K}^{-1}\varvec{W}},$$

(42)

where ${\varvec{K}}$ represents the stiffness matrix of the mechanism, ${\varvec{W}}$ represents the external force on the moving platform. The two mechanisms use the same structural parameters. Given the external load W is:

$${\varvec{W = }}\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {100{\kern 1pt} \, \text{N}} & { - 100{\kern 1pt} \, \text{N}} & { - 100{\kern 1pt} \, \text{N}} \\ \end{array} } & {10{\kern 1pt} \, \text{N} \cdot {\text{m}} } & { - 10 \, \text{N} \cdot {\text{m}} } & {10{\kern 1pt} \, \text{N} \cdot {\text{m}} } \\ \end{array} } \right]^{\text{T}} .$$

(43)

When n=2, the same cross-sectional parameters of the z axis (${{z}} = 0.45 \, \text{m}$) are selected, and the VM index comparison diagram of the two mechanisms in the working space is shown in Figure 8.

Figure 8

VM index of 3-P2(RPR) PM and 3-PRPR PM

Figure 8 shows that compared with the coupling mechanism 3-PRPR, the VM index of the new decoupled mechanism 3-P2(RPR) is significantly improved. The VM value of 3-P2(RPR) is above the value of 3-PRPR at any position, which indicates that 3-P2(RPR) has less deformation and greater stiffness under the same load. According to the rigidity requirement, the rigidity characteristic of the mechanism can be improved by adjusting the number of n, and the precision of the mechanism will be improved as well.

As shown in Figures 9 and 10, through ANSYS finite element analysis software, the stiffness of 3-P2 (RPR) PM and typical 3-CPR coupled PM were compared. The two mechanisms are selected the same structural parameters and given the same boundary conditions (7500 N load along the z direction). The maximum deformation of 3CPR PM and 3-P2 (RPR) PM is 5.91 mm and 1.44 mm respectively. The addition of closed loop makes the stiffness performance of 3-P2 (RPR) PM much better than that of 3CPR coupled PM.

Figure 9

Deformation cloud of 3-CPR coupled PM

Figure 10

Deformation cloud of 3-PRPR PM

6.4 Application Prospect of Decoupled Parallel Mechanism with High Stiffness

For the sake of the accurate fatigue life prediction of a car, there is often a durability testing before the car leaving the factory. The test usually carried out by a wheel coupled durability test platform. However, the platform can only simulate the excitation in the vertical direction, and ignore the actual excitation in the lateral and tangential directions, resulting in low road spectrum reproduction accuracy. Since the road map is very complex, the application of the highly coupled PM will increase the difficulty of control and cannot meet the requirements of road map reproduction accuracy. Therefore, a decoupled PM can be used to solve this problem [38].

However, the existing fully decoupled PMs have poor rigidity and cannot be applied to high-precision automobile durability testing. The high-stiffness PM with closed-loop units synthesized in this paper, as shown in Figure 6, has the characteristics of decoupled and high stiffness, which can provide lateral and tangential excitation for durability testing, and improve the accuracy of road spectrum reproduction. It can also reduce the difficulty of control, so that it has a better application prospect in automobile durability testing.

7 Conclusions

• 1. A new synthesis method of a fully decoupled 3T PM with closed-loop units and high rigidity was proposed.

• 2. Aiming at the problem of poor stiffness of decoupled PMs, a closed-loop unit construction method was developed, and more than 50 new configurations of high-rigidity fully decoupled 3T PMs with closed-loop units were synthesized. The structural form of the closed-loop unit is geometrically closed. The unit is composed of two completely consistent limbs, which are driven simultaneously.

• 3. The kinematics and stiffness analyses verify the correctness of the 3T PM with fully decoupled and high rigidity characteristic. The addition of the closed-loop units makes the stiffness performance of the orthogonal decoupled PM better than that of the orthogonal coupled PM and typical 3-CPR coupled PM. This paper provided new ideas for the synthesis of the high stiffness and fully decoupled PMs with closed-loop units for automobile durability testing.