Type Synthesis of 1T2R Parallel Mechanisms Using Structure Coupling-Reducing Method

Abstract

Direct kinematics with analytic solutions is critical to the real-time control of parallel mechanisms. Therefore, the type synthesis of a mechanism having explicit form of forward kinematics has become a topic of interest. Based on this purpose, this paper deals with the type synthesis of 1T2R parallel mechanisms by investigating the topological structure coupling-reducing of the 3UPS&UP parallel mechanism. With the aid of the theory of mechanism topology, the analysis of the topological characteristics of the 3UPS&UP parallel mechanism is presented, which shows that there are highly coupled motions and constraints amongst the limbs of the mechanism. Three methods for structure coupling-reducing of the 3UPS&UP parallel mechanism are proposed, resulting in eight new types of 1T2R parallel mechanisms with one or zero coupling degree. One obtained parallel mechanism is taken as an example to demonstrate that a mechanism with zero coupling degree has an explicit form for forward kinematics. The process of type synthesis is in the order of permutation and combination; therefore, there are no omissions. This method is also applicable to other configurations, and novel topological structures having simple forward kinematics can be obtained from an original mechanism via this method.

1 Introduction

Forward kinematics, which is one of the fundamental issues in the kinematic analysis of a parallel mechanism (PM) [1], refers to evaluating the pose of a platform from a set of specified values of actuated joint parameters. Generally, the forward kinematic analysis of a PM is much more complex than its reverse process, i.e., inverse kinematics. The difficulty is mainly in relation to the dependent motions of the passive joints in the mechanism.

For a better illustration, three planar open kinematic chains that can be employed to compose planar PMs are given as examples (see Figure 1). The first one is a 2-degree-of-freedom (DOF) planar open kinematic chain applying a constraint on a moving platform. It is easy to see that given the position of point A, the lengths of the linkages, and input angle θ, the position of point D on the moving platform can be directly determined using the vector pointing from point A to point D. Meanwhile, for the second and third cases, because of the need to evaluate passive joint parameters α* and β*, the forward kinematics of point D cannot be solved using only the kinematic equations formulated by the open kinematic chain itself, and additional equations depicting the coupled motions among the chains of the PM have to be derived. Therefore, the higher the number of passive joints in a PM, the higher the complexity of the forward kinematic analysis. In order to calculate the forward kinematic problem of a PM efficiently, great efforts have been made to develop numerical algorithms for particular case studies [2,3,4,5,6,7,8]. Although the real-time performance benefits of these methods have been exemplified, only approximate solutions for forward kinematics have so far been attained.

Figure 1

Open kinematic chains

Because this issue is critical to the real-time control of PMs [9], the type synthesis of a mechanism having an explicit form for forward kinematics has become a topic of interest. However, the majority of currently known type syntheses aim to realize output motion only [10,11,12]. To design PMs with analytic direct kinematics, a metric for assessing the complexity of forward kinematics at a topological level has to first be defined. For this purpose, the motion of coupling degree, which is denoted by κ, is proposed in the theory of mechanism topology [13, 14], giving the minimum number of passive joint parameters that need to be solved in the process of forward kinematic analysis. For a specific PM, the value of κ is usually an integral number larger than zero. However, when its value is equal to zero, analytical solutions for forward kinematics can be achieved. Through further examination of the coupling degree of a PM, the approach known as structure coupling-reducing (SCR), which aims to reduce the coupling degree while keeping the DOF and position and orientation characteristics (POC) [14] of the mechanism unchanged, is developed in [15, 16]. In this manner, new topological structures that have simple forward kinematics can be obtained from an original mechanism by rearranging the axes and positions of the joints. With the aid of this approach, eight new 3-DOF translational PMs [17], three types of 3T1R PMs [18], and sixteen novel 6-DOF PMs [19,20,21,22,23] have been synthesized.

Drawing on the SCR method, this paper deals with the type synthesis of 1T2R PMs by reducing the coupling degree of the 3UPS&UP PM within the Tricept robot [24,25,26]. The rest of this paper is organized as follows. In Section 2, the topological characteristics of the 3UPS&UP PM are systemically analyzed, followed by the calculation of the coupling degree of the mechanism. In Section 3, the SCR methods are then introduced to synthesize 1T2R PMs with new topologies. Finally, in Section 4, the forward kinematic analysis of one of the obtained PMs is performed to demonstrate the validity of the method, before conclusions are drawn in Section 5.

2 Topological Characteristics Analysis

In this section, the topological characteristics [13, 14] in terms of the POC set, decomposition of single-open chains (SOC), constraint degree Δ, and coupling degree κ of the 3UPS&UP PM within the Tricept robot are systematically analyzed.

2.1 Mobility Analysis

As shown in Figure 2, the 3UPS&UP PM consists of three active limbs and one passive limb connecting the base to the moving platform. The active limb i ($i = 1,2,3$) is composed of a universal joint U_i, a prismatic joint P_i, and a spherical joint S_i. Meanwhile, the passive limb is composed of a universal joint U₄ and a prismatic joint P₄.

Figure 2

Schematic diagram of 3UPS&UP PM

Based on the theory of mechanism topology [14], the POC sets indicating the motion characteristics of the U, P, and S joints (see Figure 3) can be expressed as follows:

$$\begin{aligned} &M_{{{\text{U}}i}} = \left[ \begin{array}{*{20}l} & \left\{ {t^{2} ( \bot {\varvec{\uprho}}_{{{\text{U}}i}} )} \right\} \\ & r^{2} (||{\text{U}}_{i} ) \\ \end{array} \right],\;M_{{{\text{P}}i}} = \left[ \begin{array}{*{20}l} & t^{1} (||{\text{P}}_{i} ) \\ & r^{0} \\ \end{array} \right],\;\\ &M_{{{\text{S}}i}} = \left[ \begin{array}{*{20}l} &\left\{ {t^{2} (\bot {\varvec{\uprho}}_{{{\text{S}}i}} )} \right\} \\ & r^{3} \\ \end{array} \right], \end{aligned}$$

(1)

where t and r represent translation and rotation, respectively; ${\varvec{\uprho}}_{{{\text{U}}i}}$ and ${\varvec{\uprho}}_{{{\text{S}}i}}$ are the vectors from the centers of U_i and S_i to a reference point O, respectively; the superscript is the number of translational/rotational motion; (*) describes the direction of translation or the axis of rotation; while {*} denotes the parasitic translations induced by rotations. For example, $t^{1} (||{\text{P}}_{i} )$ and r⁰ in M_Pi indicate that a prismatic joint P_i can achieve only one translational motion parallel to the direction of the joint.

Figure 3

Motion characteristics of U_i, P_i, and S_i joints

The joints are serially connected in a limb, and therefore, the POC set of a limb can be expressed as the union of the POC sets of all joints. Here, the operation rules of “union” are presented in Refs. [27, 28]. The POC sets of UPS and UP limbs can then be obtained via

$$\begin{aligned} M_{bi} & = M_{{{\text{U}}i}} \cup M_{{{\text{P}}i}} \cup M_{{{\text{S}}i}} \\ & = \left[ \begin{array}{*{20}l} \left\{ {t^{2} ( \bot {\varvec{\uprho}}_{{{\text{U}}i}} )} \right\} \\ r^{2} (||{\text{U}}_{i} ) \\ \end{array} \right] \cup \left[ \begin{array}{*{20}l} t^{1} (||{\text{P}}_{i} )\\ r^{0} \\ \end{array} \right] \cup \left[ \begin{array}{*{20}l} \left\{ {t^{2} (\bot {\varvec{\uprho}}_{{{\text{S}}i}} )} \right\}\\ r^{3} \\ \end{array} \right] \\ & = \left[ \begin{array}{*{20}l} t^{1} (||{\text{P}}_{i} ) \\ r^{0} \\ \end{array} \right] \cup \left[ \begin{array}{*{20}l} \{ t^{2} ( \bot {\varvec{\uprho}}_{{{\text{U}}i}} )\} \cup \{t^{2} (\bot {\varvec{\uprho}}_{{{\text{S}}i}} )\} \\ r^{2} (||{\text{U}}_{i} ) \cup r^{3} \\ \end{array} \right] \\ & = \left[ \begin{array}{*{20}l} t^{1} (||{\text{P}}_{i} ) \cup t^{2} ( \bot {\varvec{\uprho}}_{{{\text{U}}i}} ) \\ r^{3} \\ \end{array} \right] = \left[ \begin{array}{*{20}l} t^{3} \\ r^{3} \\ \end{array} \right],\quad i = 1,2,3, \\ \end{aligned}$$

(2)

$$\begin{aligned} M_{b4} & = M_{\text{U4}} \cup M_{\text{P4}} = \left[ \begin{aligned} &\{ t^{2} ( \bot {\varvec{\uprho}}_{{{\text{U}}4}} )\} \\ & r^{2} (||{\text{U}}_{4} ) \\ \end{aligned} \right] \cup \left[ \begin{aligned} &t^{1} ( | | {\text{P}}_{4} ) \\ & r^{0} \\ \end{aligned} \right] \\ & = \left[ \begin{array}{*{20}l} t^{1} (||{\text{P}}_{4} ) \cup \{ t^{2} ( \bot {\varvec{\uprho}}_{{{\text{U}}4}} )\} \\ r^{2} (||{\text{U}}_{4} ) \\ \end{array} \right]. \\ \end{aligned}$$

(3)

In Eq. (2), because the rotational axes of S_i could be parallel to those of U_i, the union of $\{ t^{2} ( \bot {\varvec{\uprho}}_{{{\text{U}}i}} )\}$ and $\{ t^{2} (\bot {\varvec{\uprho}}_{{{\text{S}}i}})\}$ leads to $t^{2} ( \bot {\varvec{\uprho}}_{{{\text{U}}i}} )$, according to the rules given in Ref. [21]. Furthermore, because the directions of $t^{2} ( \bot {\varvec{\uprho}}_{{{\text{U}}i}} )$ are perpendicular to the direction of P_i, the union of $t^{2} ( \bot {\varvec{\uprho}}_{i} )$ and $t^{1} (||{\text{P}}_{i} )$ results in $t^{3}$. In Eq. (3), the translational directions of $\{ t^{2} ( \bot {\varvec{\uprho}}_{\text{U4}})\}$ and $t^{1} (||{\text{P}}_{4} )$ are independent; therefore, the union of these two translational motions is $t^{1} (||{\text{P}}_{4} ) \cup \{ t^{2} ( \bot {\varvec{\uprho}}_{{{\text{U}}4}} )\}$. From Eqs. (2) and (3), it can be found that the UPS limb is a 6-DOF unconstrained open kinematic chains, while the UP limb has one translational and two rotational independent motions.

Because of the parallel topology of the mechanism, the POC set of the moving platform, M_pa, can be obtained by taking the intersection of the POC sets of the limbs. Based on the operation rules of “intersection” given in Refs. [27, 28], the POC set is defined as follows:

$$\begin{aligned} M_{pa} & = \mathop \cap \limits_{i = 1}^{4} M_{bi} \\ & = \left[ \begin{array}{*{20}l} t^{1} (||{\text{P}}_{4} ) \cup \{ t^{2} ( \bot {\text{U}}_{4} )\} \\ r^{2} (||{\text{U}}_{4} ) \\ \end{array} \right] \cap \left[ \begin{aligned} t^{3} \\ r^{3} \\ \end{aligned} \right] \cap \left[ \begin{aligned} t^{3} \\ r^{3} \\ \end{aligned} \right] \cap \left[ \begin{aligned} t^{3} \\ r^{3} \\ \end{aligned} \right] \\ & = \left[ \begin{array}{*{20}l} t^{1} (||{\text{P}}_{4} ) \cup \{ t^{2} ( \bot {\text{U}}_{4} )\} \\ r^{2} (||{\text{U}}_{4} ) \\ \end{array} \right], \\ \end{aligned}$$

(4)

from which it can be seen that M_pa is identical with M_b4. For this reason, the UP limb is also known as the POC chain [29].

2.2 Coupling Degree

The coupling degree can be considered as a metric for assessing the complexity of forward kinematics. For the evaluation of coupling degree of the PM, the mechanism must be decomposed into several SOCs and independent loops [28]. As shown in Figure 4, the decomposition is given as follows:

$$\begin{aligned} & {\text{SOC}}_{ 1} \{ {\text{U}}_{ 1} - {\text{P}}_{ 1} - {\text{S}}_{ 1} - {\text{P}}_{ 4} - {\text{U}}_{ 4} \} , \\ & {\text{SOC}}_{ 2} \{ {\text{U}}_{ 2} - {\text{P}}_{ 2} - {\text{S}}_{ 2} \} ,{\text{ SOC}}_{ 3} \{ {\text{U}}_{ 3} - {\text{P}}_{ 3} - {\text{S}}_{ 3} \} . \\ \end{aligned}$$

Figure 4

Decomposition of the 3UPS&UP PM

The first independent loop, Loop1, can then be defined by fixing both end links of SOC₁ according to Ref. [14], while the second independent loop, Loop2, is obtained by taking Loop1 as a sub-mechanism and attaching two end links of SOC₂ to Loop1. Analogously, the third independent loop, Loop3, is defined by attaching two end links of SOC₃ to Loop2 (see Figure 4). With these definitions, the number of independent displacement equations of the three loops can be computed following the method presented in Ref. [28].

$$\begin{aligned} \xi_{L1} & = \dim \left\{ {M_{{b_{1} }} \cup M_{b4} } \right\} \\ & = \dim \left\{ {\left[ \begin{aligned}& t^{1} (||{\text{P}}_{4} ) \cup \{ t^{2} ( \bot {\text{U}}_{4} )\} \\ &r^{2} (||{\text{U}}_{4} ) \\ \end{aligned} \right] \cup \left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right]} \right\} \\ & = \dim \left\{ {\left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right]} \right\} = 3 + 3 = 6, \\ \end{aligned}$$

(5)

$$\begin{aligned} \xi_{L2} & = \dim \left\{ {(M_{{b_{1} }} \cap M_{b4} ) \cup M_{b2} } \right\} \\ & = \dim \left\{ {\left( {\left[ \begin{array}{*{20}l} &t^{1} (||{\text{P}}_{4} ) \cup \{ t^{2} ( \bot {\text{U}}_{4} )\} \\ & r^{2} (||{\text{U}}_{4} ) \\ \end{array} \right] \cap \left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right]} \right) \cup \left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right]} \right\} \\ & = \dim \left\{ {\left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right]} \right\} = 3 + 3 = 6, \\ \end{aligned}$$

(6)

$$\begin{aligned} \begin{aligned} \xi_{L3} & = \dim \left\{ {(M_{{b_{1} }} \cap M_{b4} \cap M_{b2} ) \cup M_{b3} } \right\} \\ & = \dim \left\{ {\left( {\left[ \begin{array}{*{20}l} & t^{1} (||{\text{P}}_{4} ) \cup \{ t^{2} ( \bot {\text{U}}_{4} )\} \\ & r^{2} (||{\text{U}}_{4} ) \\ \end{array} \right]}\right.}\right. \\&\quad \left.{\left.{\cap \left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right] \cap \left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right]} \right) \cup \left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right]} \right\} \\ & = \dim \left\{ {\left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right]} \right\} = 3 + 3 = 6, \\ \end{aligned} \end{aligned}$$

(7)

where dim{*} denotes the dimension, i.e., the number of independent motions, of a POC set. Subsequently, to evaluate the coupling degree, the constraint degree describing the constraint relationship of a SOC has to be calculated [14]. For this specific case, the constraint degrees of the three SOCs are

$${\Delta}_{1} = \sum\limits_{i = 1}^{5} {f_{i} } - I_{1} - \xi_{{L_{1} }} = 9 - 1 - 6 = 2,$$

$${\Delta}_{2} = \sum\limits_{i = 1}^{3} {f_{i} } - I_{2} - \xi_{{L_{2} }} = 6 - 1 - 6 = - 1,$$

$${\Delta}_{3} = \sum\limits_{i = 1}^{3} {f_{i} } - I_{3} - \xi_{{L_{3} }} = 6 - 1 - 6 = - 1,$$

where f_i is the DOF of the ith joint, and I_j is the number of actuated joints in the jth SOC ($j = 1,2,3$). The constraint degree of SOC₁ is shown to be 2, which means that two passive joint parameters have to be solved to determine the motion of SOC₁. Meanwhile, the constraint degree of SOC₂ (SOC₃) is $- 1$, indicating that a constraint equation should be established to solve one of the passive joint parameters in the kinematic equations of SOC₁. Finally, the coupling degree of the 3UPS&UP PM can be evaluated via [14]

$$\kappa = \frac{1}{2}\sum\limits_{j = 1}^{3} {|{\Delta}_{j} |} = \frac{1}{2}(2 + | - 1| + | - 1|) = 2.$$

(8)

This relationship demonstrates that the constraints of the three SOCs are highly coupled. Therefore, there is no analytic solution for the forward kinematics of the 3UPS&UP PM. Approaches for the type synthesis of new topological structures with the same topological characteristics but using simple forward kinematics are investigated in the following section.

3 Type Synthesis

Three SCR methods, which aim to reduce the coupling degree of the 3UPS&UP PM and obtain new mechanisms that have simple forward kinematics, are presented in this section.

3.1 Method One: Changing the Passive POC Chain into Active Chain

In the 3UPS&UP PM, the UP limb is the POC chain because of M_b4 being identical with M_pa. When the passive prismatic joint in the POC chain is changed into an active one (see Figure 5), the constraint degree ${\Delta}_{1}$ of SOC₁ can be reduced to

$$\begin{aligned} &{\text{SOC}}_{ 1} \left\{ {{\text{U}}_{ 1} - {\text{P}}_{ 1} - {\text{S}}_{ 1} - {\text{P}}_{ 4} - {\text{U}}_{ 4} } \right\},\;\\&{\Delta}_{1} = \sum\limits_{i = 1}^{5} {f_{i} } - I_{1} - \xi_{{L_{1} }} = 9 - 2 - 6 = 1, \end{aligned}$$

because I₁ = 2 in this equation. Furthermore, in order to achieve a PM without redundant actuation, the active joint P₃ in SOC₃ is taken as a passive joint, resulting in

$$\begin{aligned} &{\text{SOC}}_{ 3} \{ {\text{U}}_{ 3} - {\text{P}}_{ 3} - {\text{S}}_{ 3} \} ,\;\\ &{\Delta}_{3} = \sum\limits_{i = 1}^{3} {f_{i} } - I_{3} - \xi_{{L_{3} }} = 6 - 0 - 6 = 0. \end{aligned}$$

Figure 5

The first loop

Hence, the coupling degree of this new PM (No.1 (a) in Table 1) is

$$\kappa = \frac{1}{2}\sum\limits_{j = 1}^{3} {|{\Delta}_{j} |} = \frac{1}{2}(1 + | - 1| + |0|) = 1,$$

(9)

indicating that only one passive joint parameter needs to be solved in the forward kinematics. Because limb 3 becomes a 6-DOF passive limb in this special case, it has no effect on M_pa. Therefore, it is removable, leading to the PM (No.1 (b) in Table 1) within the TriVariant robot [30].

Table 1 Topological structure obtained via design of SCR for 3UPS&UP

3.2 Method Two: Making the Centers of Spherical Joints Coincident

Another approach to reduce the coupling degree is by making the centers of the spherical joints on the moving platform coincident. For example, the centers of spherical joints S₁ and S₃ are coincident, as shown in No. 3 in Table 1. The SOCs of the obtained mechanism (see Figure 6) and their constraint degrees can then be redefined as

$$\begin{aligned} &{\text{SOC}}_{ 1} \{ {\text{U}}_{ 1} - {\text{P}}_{ 1} - {\text{S}}_{ 1 3} - {\text{P}}_{ 3} - {\text{U}}_{ 3} \} ,\;\\&{\Delta}_{1} = \sum\limits_{i = 1}^{5} {f_{i} } - I_{1} - \xi_{{L_{1} }} = 9 - 2 - 6 = 1, \end{aligned}$$

$$\begin{aligned}& {\text{SOC}}_{ 2} \{ {\text{U}}_{ 4} - {\text{P}}_{ 4} \} ,\;\\&{\Delta}_{2} = \sum\limits_{i = 1}^{2} {f_{i} } - I_{2} - \xi_{{L_{2} }} = 3 - 3 = 0, \end{aligned}$$

$$\begin{aligned} &{\text{SOC}}_{ 3} \{ {\text{U}}_{ 2} - {\text{P}}_{ 2} - {\text{S}}_{ 2} \} ,\;\\&{\Delta}_{3} = \sum\limits_{i = 1}^{3} {f_{i} } - I_{3} - \xi_{{L_{3} }} = 6 - 1 - 6 = - 1, \end{aligned}$$

resulting in the coupling degree of this new mechanism:

Figure 6

Topological decomposition of No. 3 PM

$$\kappa = \frac{1}{2}\sum\limits_{j = 1}^{3} {|{\Delta}_{j} |} = \frac{1}{2}(|1| + |0| + | - 1|) = 1.$$

(10)

Therefore, a new type of 1T2R PM with lower coupling degree is synthesized. In addition, we can also make the centers of three spherical joints connected to the moving platform coincident. In this way, a new mechanism, as shown in No.4 in Table 1, can then be obtained. For this mechanism, the redefined SOCs and their constraint degrees are given as follows:

$$\begin{aligned} &{\text{SOC}}_{ 1} \{ {\text{U}}_{ 1} - {\text{P}}_{ 1} - {\text{S}}_{ 1 2 3} - {\text{P}}_{ 3} - {\text{U}}_{ 3} \} ,\;\\&{\Delta}_{1} = \sum\limits_{i = 1}^{5} {f_{i} } - I_{1} - \xi_{{L_{1} }} = 8 - 2 - 6 = 0, \end{aligned}$$

$${\text{SOC}}_{ 2} \{ {\text{U}}_{ 4} - {\text{P}}_{ 4} \} ,\;{\Delta}_{2} = \sum\limits_{i = 1}^{2} {f_{i} } - I_{2} - \xi_{{L_{2} }} = 3 - 3 = 0,$$

$${\text{SOC}}_{ 3} \{ {\text{U}}_{ 2} - {\text{P}}_{ 2} - {\text{R}}^{{({\text{U1}} - {\text{U3}})}} \} ,\;{\Delta}_{3} = \sum\limits_{i = 1}^{3} {f_{i} } - I_{3} - \xi_{{L_{3} }} = 4 - 1 - 3 = 0.$$

Here, the spherical joint S₂ in SOC₃ is degenerated into a revolute joint R^(U1-U3) with the rotational axis parallel to the line passing through the centers of joints U₁ and U₃. The coupling degree of this mechanism is then

$$\kappa = \frac{1}{2}\sum\limits_{j = 1}^{3} {|{\Delta}_{j} |} = \frac{1}{2}(|0| + |0| + |0|) = 0.$$

(11)

Because $\kappa = 0$, it can be concluded that the PM has an explicit form for forward kinematics.

3.3 Method Three: Integrating the Rotational Axes of Universal Joints

This method aims to directly reduce $\sum {f_{i} }$ in a SOC. Note that the joint U can be decomposed into a proximal revolute joint R′ and a distal revolute joint R with their rotational axes being perpendicular. Hence, in the 3UPS&UP PM, the revolute joints R′₂ and R′₄ of U₂ and U₄, respectively, could be made coincident and integrated into a revolute joint R₂₄ (see No. 5 in Table 1). The kinematic chains U₄P₄ and U₂P₂S₂ can then be regarded as a hybrid single-open chain, HSOC₁ [15].

$${\text{HSOC}}_{ 1} \{ {\text{R}}_{ 2 4} - ({\text{R}}_{ 2} \bot {\text{P}}_{ 2} - {\text{S}}_{ 2} ) \, ({\text{R}}_{ 4} \bot {\text{P}}_{ 4} )\}.$$

The SOCs and corresponding constraint degrees of this mechanism (see Figure 7) can then be defined as

Figure 7

Topological decomposition of No.5 PM

$$\begin{aligned} &{\text{SOC}}_{ 1} \{ {\text{HSOC}}_{ 1} - {\text{S}}_{ 3} - {\text{P}}_{ 3} - {\text{U}}_{ 3} \} ,\;\\&{\Delta}_{1} = \sum\limits_{i = 1}^{5} {f_{i} } - I_{1} - \xi_{{L_{1} }} = 9 - 2 - 6 = 1, \end{aligned}$$

$$\begin{aligned} &{\text{SOC}}_{ 2} \{ {\text{U}}_{ 1} - {\text{P}}_{ 1} - {\text{S}}_{ 1} \} ,\;\\&{\Delta}_{2} = \sum\limits_{i = 1}^{2} {f_{i} } - I_{2} - \xi_{{L_{2} }} = 6 - 1 - 6 = - 1. \end{aligned}$$

Consequently, the coupling degree of the new PM is

$$\kappa = \frac{1}{2}\sum\limits_{j = 1}^{2} {|{\Delta}_{j} |} = \frac{1}{2}(|1| + | - 1|) = 1.$$

(12)

Furthermore, R′₂, R′₃, and R′₄ of U₂, U₃, and U₄, respectively, can also be integrated into one revolute joint R₂₃₄. In this manner, the kinematic chains U₄P₄, U₃P₃S₃, and U₂P₂S₂ are converted into another HSOC₁,

$${\text{HSOC}}_{ 1} \{ {\text{R}}_{ 2 3 4} - ({\text{R}}_{ 4} \bot {\text{P}}_{ 4} )({\text{R}}_{ 2} \bot {\text{P}}_{ 2} - {\text{S}}_{ 2} )({\text{R}}_{ 3} \bot {\text{P}}_{ 3} - {\text{S}}_{ 3} )\} ,$$

resulting in the R(2RPS&RP)&UPS PM (No.6 in Table 1) within the TriMule robot [31]. For this mechanism, its SOC can be expressed as

$$\begin{aligned} &{\text{SOC}}_{ 1} \{ {\text{HSOC}}_{ 1} - {\text{S}}_{ 1} - {\text{P}}_{ 1} - {\text{U}}_{ 1} \} ,\;\\&{\Delta}_{1} = \sum\limits_{i = 1}^{5} {f_{i} } - I_{1} - \xi_{{L_{1} }} = 9 - 3 - 6 = 0. \end{aligned}$$

Because there is only one SOC, the coupling degree of this PM is zero. Moreover, if the revolute joints R′₁ and R′₃ of U₁ and U₃, respectively, and the revolute joints R′₂ and R′₄ of U₂ and U₄, respectively, are made coincident and integrated into revolute joints R₁₃ and R₂₄, respectively, a new mechanism No.7 in Table 1 can be obtained. For this case, the kinematic chains U₁P₁S₁ and U₃P₃S₃, and U₂P₂S₂ and U₄P₄ are converted into HSOC₁ and HSOC₂, respectively.

$${\text{HSOC}}_{ 1} \{ {\text{R}}_{ 1 3} - ({\text{R}}_{ 1} \bot {\text{P}}_{ 1} - {\text{S}}_{ 1} ) \, ({\text{R}}_{ 3} \bot {\text{P}}_{ 3} - {\text{S}}_{ 3} )\},$$

$${\text{HSOC}}_{ 2} \{ {\text{R}}_{ 2 4} - ({\text{R}}_{ 2} \bot {\text{P}}_{ 2} - {\text{S}}_{ 2} ) \, ({\text{R}}_{ 4} \bot {\text{P}}_{ 4} )\}.$$

The SOC of the obtained PM can be expressed as

$$\begin{aligned} &{\text{SOC}}_{ 1} \{ {\text{HSOC}}_{ 1} - {\text{HSOC}}_{ 2} \} ,\;\\&{\Delta}_{1} = \sum\limits_{i = 1}^{5} {f_{i} } - I_{1} - \xi_{{L_{1} }} = 9 - 3 - 6 = 0. \end{aligned}$$

This mechanism also has only one SOC, and its coupling degree is zero. Besides, if the revolute joints R′₂ and R′₃ of U₂ and U₃, respectively, in the PM No. 1(b) in Table 1 are integrated into one revolute joint R₂₃, another new mechanism is obtained, as shown in No. 2 in Table 1. In this way, the kinematic chains U₂P₂S₂ and U₃P₃S₃ are converted into HSOC₁:

$${\text{HSOC}}_{ 1} \{ {\text{R}}_{ 2 3} - ({\text{R}}_{ 2} \bot {\text{P}}_{ 2} - {\text{S}}_{ 2} ) \, ({\text{R}}_{ 3} \bot {\text{P}}_{ 3} - {\text{S}}_{ 3} )\} .$$

The SOC and corresponding constraint degree can be expressed as

$${\text{SOC}}_{ 1} \{ {\text{HSOC}}_{ 1} - {\text{P}}_{ 1} - {\text{U}}_{ 1} \} ,\;{\Delta}_{1} = \sum\limits_{i = 1}^{5} {f_{i} } - I_{1} - \xi_{{L_{1} }} = 9 - 3 - 6 = 0.$$

It can be proved that the coupling degree of this PM is also zero.

4 Example

In this section, the forward kinematic analysis of the obtained R(2RPS&RP)&UPS PM is given as an example to verify that a PM with $\kappa = 0$ has an explicit form for forward kinematics, which would illustrate the feasibility of the proposed type-synthesis method.

4.1 Coordinate System

The schematic diagram of the R(2RPS&RP)&UPS PM is shown in Figure 8, where points B, B₁, B₂, and B₃ are the centers of joints R₄, U₁, R₂, and R₃, respectively. Point O is the intersection of the axial axis of the passive limb R₄P₄ and its normal plane in which all centers of S joints, points A₁, A₂, and A₃, are placed. A reference frame B-XYZ is attached to point B, satisfying Y⊥B₂B₃ and Z⊥△B₁B₂B₃, while a body-fixed frame O-xyz is established at point O, satisfying y⊥A₂A₃ and z⊥△A₁A₂A₃. The orientation of O-xyz with respect to B-XYZ can then be evaluated via

$${\varvec{R}} = {\text{Rot}}\left( {X,\theta_{1} } \right){\text{Rot}}\left( {y,\theta_{2} } \right) = \left[ {\begin{array}{*{20}c} {{\text{c}}\theta_{2} } & 0 & {\text{s} \theta_{2} } \\ {\text{s} \theta_{1} \text{s} \theta_{2} } & {{\text{c}}\theta_{1} } & { - \text{s} \theta_{1} {\text{c}}\theta_{2} } \\ { - {\text{c}}\theta_{1} \text{s} \theta_{2} } & {\text{s} \theta_{1} } & {{\text{c}}\theta_{1} {\text{c}}\theta_{2} } \\ \end{array} } \right],$$

(13)

where c and s represent “cos” and “sin”, respectively.

Figure 8

Schematic diagram of R(2RPS&RP)&UPS PM

4.2 Forward Kinematics

When point O is taken as the reference point, its position vector evaluated in B-XYZ can be expressed as

$$q_{4} \varvec{s}_{4} = \varvec{b}_{i} + q_{i} \varvec{s}_{i} - \varvec{a}_{i} ,\quad i = 1,2,3,$$

(14)

where $q_{i}$ is the length, and $\varvec{s}_{i}$ is the unit vector along the axial axis of limb I ($i = 1,2,3,4$); ${\varvec{a}}_{i} = {\varvec{Ra}}_{i0}$; ${\varvec{a}}_{i0}$ and ${\varvec{b}}_{i}$ are the positioning vectors of $A_{i}$ and $B_{i}$ ($i = 1,2,3$), respectively, measured in O-xyz and B-XYZ, respectively. Noting that

$$\varvec{s}_{4} = {\varvec{R}}\left( {\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} } \right)^{\text{T}} = \left( {\begin{array}{*{20}c} {\text{s} \theta_{2} } & { - \text{s} \theta_{1} {\text{c}}\theta_{2} } & {{\text{c}}\theta_{1} {\text{c}}\theta_{2} } \\ \end{array} } \right)^{\text{T}}$$

(15)

and taking norms on both sides of rearranged Eq. (14) leads to

$$q_{i} = \left\| {q_{4} \varvec{s}_{4} + \varvec{a}_{i} - \varvec{b}_{i} } \right\|,\quad i = 1,2,3.$$

(16)

Expanding Eq. (16) then gives the following kinematic equations:

$$a_{1}^{2} + b_{1}^{2} + q_{0}^{2} - 2a_{1} b_{1} \text{c} \theta_{1} - 2b_{1} q_{0} \text{s} \theta_{1} \text{c} \theta_{2} - q_{1}^{2} = 0,$$

(17)

$$a_{2}^{2} + b_{2}^{2} + q_{0}^{2} - 2a_{2} b_{2} \text{c} \theta_{2} - 2b_{2} q_{0} \text{s} \theta_{2} - q_{2}^{2} = 0,$$

(18)

$$a_{2}^{2} + b_{2}^{2} + q_{0}^{2} - 2a_{2} b_{2} \text{c} \theta_{2} + 2b_{2} q_{0} \text{s} \theta_{2} - q_{3}^{2} = 0,$$

(19)

where $a_{i} = \left\| {\varvec{a}_{i} } \right\|$ and $b_{i} = \left\| {\varvec{b}_{i} } \right\|$ ($i = 1,2,3$), satisfying $a_{2} = a_{3}$ and $b_{2} = b_{3}$. Combining Eq. (18) with Eq. (19) results in

$$\left\{ \begin{array}{*{20}l} \text{c} \theta_{2} = \frac{{2\left( {a_{2}^{2} + b_{2}^{2} + q_{0}^{2} } \right) - q_{2}^{2} - q_{3}^{2} }}{{4a_{2} b_{2} }}, \hfill \\ \text{s} \theta_{2} = \frac{{q_{3}^{2} - q_{2}^{2} }}{{4b_{2} q_{0} }}. \hfill \\ \end{array} \right.$$

(20)

Considering that s²θ₂ +c²θ₂=1, we can get

$$\left( {q_{0}^{2} } \right)^{3} + P_{1} \left( {q_{0}^{2} } \right)^{2} + P_{2} q_{0}^{2} + P_{3} = 0,$$

(21)

$$P_{1} = 2a_{2}^{2} + 2b_{2}^{2} - q_{2}^{2} - q_{3}^{2} ,$$

$$P_{2} = {{P_{1}^{2} } \mathord{\left/ {\vphantom {{P_{1}^{2} } 4}} \right. \kern-0pt} 4} - 4a_{2}^{2} b_{2}^{2} ,\quad P_{3} = {{a_{2}^{2} \left( {q_{3}^{2} - q_{2}^{2} } \right)^{2} } \mathord{\left/ {\vphantom {{a_{2}^{2} \left( {q_{3}^{2} - q_{2}^{2} } \right)^{2} } 4}} \right. \kern-0pt} 4}.$$

When Eq. (21) is solved, the following equations can be obtained:

$$q_{0} = \sqrt {\sqrt[3]{{C_{1} }} + \sqrt[3]{{C_{2} }} - {{P_{1} } \mathord{\left/ {\vphantom {{P_{1} } 3}} \right. \kern-0pt} 3}} ,$$

(22)

$$C_{1} = - \frac{{D_{2} }}{2} + \sqrt {\left( {\frac{{D_{2} }}{2}} \right)^{2} + \left( {\frac{{D_{1} }}{3}} \right)^{3} } ,$$

$$C_{2} = - \frac{{D_{2} }}{2} - \sqrt {\left( {\frac{{D_{2} }}{2}} \right)^{2} + \left( {\frac{{D_{1} }}{3}} \right)^{3} } ,$$

$$D_{1} = P_{2} - {{P_{1}^{2} } \mathord{\left/ {\vphantom {{P_{1}^{2} } 3}} \right. \kern-0pt} 3},\quad D_{2} = P_{3} + {{2P_{1}^{3} } \mathord{\left/ {\vphantom {{2P_{1}^{3} } {27}}} \right. \kern-0pt} {27}} - {{P_{1} P_{2} } \mathord{\left/ {\vphantom {{P_{1} P_{2} } 3}} \right. \kern-0pt} 3}.$$

Substituting Eq. (22) into Eq. (20) then leads to

$$\theta_{2} {\text{ = arcsin}}\frac{{q_{3}^{2} - q_{2}^{2} }}{{4b_{2} q_{0} }},$$

(23)

$$\theta_{1} = 2\arctan \left( {\frac{{ - P_{4} - \sqrt {P_{4}^{2} - P_{6}^{2} + P_{5}^{2} } }}{{P_{6} - P_{5} }}} \right),$$

(24)

$$P_{4} = - 2b_{1} q_{0} \text{c} \theta_{2} ,\;P_{5} = - 2a_{1} b_{1} ,\;P_{6} = a_{1}^{2} + b_{1}^{2} + q_{0}^{2} - q_{1}^{2} .$$

Therefore, the R(2RPS&RP)&UPS PM is proven to have an explicit form for forward kinematics.

5 Conclusions

Investigation on the type synthesis of 1T2R PMs using the method of structure coupling-reducing is presented in this paper. The following conclusions are drawn.

• (1) Conducting a topological characteristics analysis of the 3UPS&UP PM shows that the motions of the three SOCs of the mechanism are highly coupled. Therefore, for this setup, explicit solutions for forward kinematics cannot be achieved.

• (2) Aiming at reducing the coupling degree of the 3UPS&UP PM and synthesizing new mechanisms that have simple forward kinematics, three different methods are proposed, by which eight new PMs having lower coupling degrees are obtained.

• (3) The forward kinematic analysis of the R(2RPS& RP)&UPS PM is presented. The analysis verifies that the obtained new mechanism, which has a zero coupling degree, uses an explicit form for forward kinematics.