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Abstract

The novel design of parallel mechanisms plays a key role in the potential application of parallel mechanisms. In this paper, the type synthesis of parallel mechanisms with the first class G_F sets and two-dimensional rotations is studied. The rule of two-dimensional rotations is given, which lays the theoretical foundation for the intersection operations of specific G_F sets. Next, kinematic limbs with specific characteristics are designed according to the 2-D and 3-D axes movement theorems. Finally, several synthesized parallel mechanisms with the first class G_F sets and two-dimensional rotations are illustrated to show the effectiveness of the proposed methodology.

Keywords

type synthesis parallel mechanisms generalized function set two-dimensional rotations

1. Introduction

The demand for proposing parallel mechanisms with specific characteristics has become increasingly important since parallel mechanisms have potential applications in many sectors, such as machining, positioning and manipulating. In particular, Altuzarra et al. [1] proposed some parallel mechanisms with two-dimensional rotations, including 3-, 4- and 5-DOF parallel mechanisms which can be employed in the application of machining. Up until now, a number of parallel mechanisms with two-dimensional rotations have been proposed. For instance, Gogu [2] presented several fully isotropic parallel mechanisms with four degrees of freedom of the T2R2-type. Moreover, Piccin et al. [3], Motevalli et al. [4] and Gogu [2] developed numerous parallel mechanisms with 5 DOFs and two rotations. The type synthesis of multi-degree of freedom parallel robotic mechanisms has been a hot subject of research over the last few decades. The mathematical tools employed include graph theory [5], lie group theory [6 –9], screw theory [10 –12], linear transformation [2,13], position and orientation characteristics equations [14] and G_F set theory [15,16]. A large number of parallel mechanisms have been synthesized systematically according to the above-mentioned theories, from which both industry and academia benefit.

The synthesizing method utilized in this paper does not rely on any reference coordinate system and only requires the intersection operations of sets. Moreover, the mobility of the synthesized parallel mechanisms according to G_F set theory is not instantaneous. G_F sets are introduced for the evaluation of the characteristics of the end-effectors of robotic manipulators. Moreover, the method for the type synthesis of parallel mechanisms with the first class G_F sets and two-dimensional rotations (FG2RPM for short) is studied. Furthermore, the rule of two-dimensional rotations was proposed to establish the computing algorithms of G_F sets for the type synthesis of FG2RPM. Several parallel mechanisms were depicted to show the effectiveness of the synthesizing methodology.

2. Relevant results of G_F sets theory

Many criteria have been established for evaluating the performance of robotic mechanisms. They include workspace, velocity, acceleration, payload and stiffness. It is necessary to develop an effective, efficient and general performance criterion that is applicable to the design of the topologies for robotic mechanisms. Such a general criterion should be non-algebraic, dimensionless and independent of the choice of coordinate systems. This is in contrast to the existing criteria of workspace, velocity, acceleration, payload and stiffness. The latter criteria, all of which are in algebraic form, dimensional, and related to coordinate systems, cannot handle the extremely diverse range of topology performances of robotic mechanisms. As a result, we employ a specialized set theory to describe the topological constraints that limit the kinematic mobility of the end-effectors of robotic mechanisms, which is called the generalized function set (or G_F set) [15,16].

G_F sets features the kinematic structure $G_{F} (T_{a} T_{b} T_{c}; R_{α} R_{β} R_{γ})$ , where the first elements $T_{i}, i = a, b, c$ denote the translational characteristics of the end-effector and the last elements $R_{j}, j = α, β, γ$ represent the rotational characteristics of the end-effector described in terms of three successive rotations. The axes of these three rotations are denoted by $R_{α}$ , $R_{β}$ and $R_{γ}$ , respectively, where $R_{α}$ is a base rotation axis, $R_{β}$ is the middle rotation axis relative to $R_{α}$ , and $R_{γ}$ is the last rotation axis relative to $R_{β}$ . Successive rotations are used because in the natural world, there is no occurrence of pure rotation around two fixed axes. Here, the term “pure” means with no accompanying translations. Consequently, the type synthesis of parallel mechanisms with two rotations is difficult to deal with. Moreover, when the element is non-zero, the corresponding capability exists; when the element is zero, the corresponding capability does not exist. As different points of an end-effector have different characteristics with respect to rotations, it is important to note the specific point of the end-effectors here. The characteristics of a point P on the end-effector include translational elements and rotational elements. $T_{a}$ , $T_{b}$ and $T_{c}$ are non-coplanar simultaneously while $R_{α}$ , $R_{β}$ and $R_{γ}$ always intersect at a common point P and are not coplanar simultaneously.

From the viewpoint of the influence between translation and rotation, we classify the G_F sets into two categories, namely, the first class G_F sets and the second class G_F sets. Detailed classifications of the G_F sets are listed in Table 1. Unlike the more usually used notations used to describe the characteristics of robotic manipulators (such as 3T2R, 3R2T and 2T1R), by using the concept of G_F sets we can distinguish different robotic manipulators in a simple and clear manner.

In this paper, we focus on the type synthesis of FG2RPM, indicating that the G_F sets $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0; & R_{α} & R_{β} & 0 \end{matrix})$ , $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0; & R_{α} & R_{β} & 0 \end{matrix})$ and $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c}; & R_{α} & R_{β} & 0 \end{matrix})$ will be discussed.

Table 1.

Enumeration of two classes of G_F sets

Dimension	First class G_F sets	Second class G_F sets
6	G_F^\|(T_a T_b T_c; R_α R_β R_γ
5	G_F^\|(T_a T_b T_c; R_α R_β 0)
	G_F^\|(T_a T_b 0; R_α R_β R_γ)	G_F^‖ (R_α R_β R_γ; T_a T_b 0)
4	G_F^\| (T_a T_b T_c; R_α 0 0)
	G_F^\|(T_a T_b 0; R_α R_β 0)	G_F^‖ (R_α R_β 0; T_a T_b 0)
	G_F^\|(T_a 0 0; R_α R_β R_γ)	G_F^‖(R_α R_β R_γ; T_a 0 0)
3	G_F^\|(T_a T_b T_c; 0 0 0)	G_F^‖ (R_α R_β R_γ; 0 0 0)
	G_F^\|(T_a T_b 0; R_α 0 0)	G_F^‖(R_α R_β 0; T_a 0 0)
	G_F^\|(T_a 0 0; R_α R_β 0)	G_F^‖(R_α 0 0; T_a T_b 0) ^a)
2	G_F^‖(T_a T_b 0; 0 0 0)	G_F^‖(R_α R_β 0; 0 0 0)
	G_F^\|(T_a 0 0; R_α 0 0)^b)	G_F^‖(R_α 0 0; T_a 0 0)
1	G_F^\|(T_a 0 0; 0 0 0)	G_F^‖(R_α 0 0; 0 0 0)

The G_F set whose rotation axis R_α is perpendicular to the plane containing T_a and T_b belongs to the first class G_F set G_F^‖(T_a T_b 0; R_α 0 0).

The G_F set whose rotation axis R_α coincides with the translation direction T_a belongs to the second class GF set.

3. Rule of two-dimensional rotations

Robotic manipulators may have serial or parallel topologies. While the characteristics of the end-effectors of a serial topology depend directly on the sum of the characteristics of all the kinematic pairs, the characteristics of the end-effector of a parallel topology are the intersections of the characteristics of the end-effectors of all the limbs. The existence of translations is straightforward to verify, while the existence of rotations is a little bit difficult. As such, it is necessary to find the requirements for the two dimensional rotations of rigid bodies, which are as follows.

The rule of two-dimensional rotations: when a rigid body rotates around two intersecting axes passing point A, all the vectors fixed to the rigid body should have the same rotations as the vectors fixed to point A. Moreover, point B should be able to translate freely in 3-D space in order to compensate the accompanying translations caused by the rotations around point A.

It is worth pointing out that the accompany translations are not necessary, given the fact that the two axes passing point A coincide with the two axes passing point B. Furthermore, if there is only one axis passing point A which coincides with one axis passing point B, then the translations required reduce to two dimensional translations which are perpendicular to the other axis which does not coincide.

4. Intersection algorithms of G_F sets

Utilizing the rule of two dimensional rotations, we can achieve the intersection algorithms for the three required G_F sets, which are $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0; & R_{α} & R_{β} & 0 \end{matrix})$ , $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0; & R_{α} & R_{β} & 0 \end{matrix})$ and $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c}; & R_{α} & R_{β} & 0 \end{matrix})$ respectively.

4.1. The Intersection Algorithms for the G_FF Set $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

Table 2 lists 10 possible intersection algorithms for the G_F set $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ . In particular, for the first case listed in Table 2, the required accompanying translations of the second G_F set are not necessary, since the two pairs of axes coincided. The same phenomenon holds for all those cases in Table 2 that the second G_F set does not have 3 dimensional translations. It is easy to see that all 10 cases in Table 2 obey the rule of two-dimensional rotations.

4.2. The Intersection Algorithms for the G_FF Set $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

Five possible intersection algorithms for the G_F set $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ are listed in Table 3 according to the rule of two-dimensional rotations. For the last case listed in Table 3, the two-dimensional rotation centres do not coincide with each other and the two translational planes are parallel to each other. Consequently, two-dimensional rotations exist, including the first one-dimensional rotation whose axis passing the two rotation centres and the second one-dimensional rotation whose axis is perpendicular to the translational plane. We can see that all of the 5 cases listed in Table 3 fulfil the requirements of the rule of two-dimensional rotations.

4.3. The Intersection Algorithms for the G_FF Set $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c} \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

As far as the intersection algorithms for the G_F set $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c} \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ are concerned, only two possible cases exist, which are listed in Table 4. It is easy to see that by using the first case listed in Table 4, we may synthesize parallel mechanisms with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c} \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ with symmetrical structures.

Table 2.

Intersection algorithms for $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

Table 3.

Intersection algorithms for $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

Table 4.

Intersection algorithms for $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c} \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

Intersection of GF sets	Condition	NO.
G^\|_F1 (T_a1 T_b1 T_c1; R_α1 R_β1 0)∩G^\|_F2(T_a2 T_b2 T_c2; R_α2 R_β2 0)	R_α1‖R_α2,R_β1‖R_β2	3.1
G^\|_F1 (T_a1 T_b1 T_c1; R_α1 R_β1 0)∩G^\|_F2(T_a2 T_b2 T_c2; R_α2 R_β2 R_γ2)	No requirements	3.2

5. Compositions of the characteristics of limbs for the FG2RPM

The characteristics of the end-effectors of parallel topologies are the intersections of the characteristics of multiple limbs, namely:

G_{F}^{​} = G_{F 1}^{​} \cap G_{F 2}^{​} \cap \dots \cap G_{F n}^{​}

(1)

The relationship between the number of limbs (n), the dimension of the characteristics of the end-effectors for parallel topologies(F_D), the number of actuators of limb (q_i) and the number of passive limbs (p) is as follows:

{\begin{matrix} F_{D} - \sum_{i = 1}^{n} q_{i} = 0 \\ N = F_{D} - \sum_{i = 1}^{n} (q_{i} - 1) + p \\ N = n + p \\ n \leq F_{D} \\ q_{i} \leq F_{D} (i = 1, 2, \dots, n) \end{matrix}

(2)

According to the algorithms listed in Table 2–4 and Eq. 1 and Eq. 2, we can achieve the typical compositions for the G_F sets $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ , $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ and $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c} \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ , which are listed in Table 5, Table 6 and Table 7, respectively. It is easy to see that both the F_D and n are equal to 3, 4 and 5 for the compositions listed in Table 5, Table 6 and Table 7, respectively. The values of the parameters used in Eq. 2 are as follows for all of the cases listed in Table 5–7.

\begin{array}{l} q_{i} = 1 \\ p = 0 \end{array}

(3)

It is worth pointing out that more compositions for the three required G_F sets could be achieved by using the results listed in Table 2, Table 3, Table 4, Eq. 1 and Eq. 2.

Table 5.

Typical compositions for the G_F Set $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

Compositions	Condition NO.
$\cap_{i = 1}^{3} G_{F i}^{\|} (T_{a i} T_{b i} 0; R_{α i} R_{β i} R_{γ i})$	2.5,1.10
$G_{F i}^{\|} (T_{a 1} 0 0; R_{α 1} R_{β 1} 0) \cap_{i = 2}^{3} G_{F i}^{\|} (T_{a i} T_{b i} 0; R_{α i} R_{β i} R_{γ i})$	1.4,1.4
$G_{F i}^{\|} (T_{a 1} 0 0; R_{α 1} R_{β 1} 0) \cap_{i = 2}^{3} G_{F i}^{\|} (T_{a i} T_{b i} T_{c i}; R_{α i} R_{β i} 0)$	1.5,1.5
$G_{F i}^{\|} (T_{a 1} 0 0; R_{α 1} R_{β 1} R_{γ i}) \cap_{i = 2}^{3} G_{F i}^{\|} (T_{a i} T_{b i} 0; R_{α i} R_{β i} R_{γ i})$	2.5,1.7
$G_{F i}^{\|} (T_{a 1} 0 0; R_{α 1} R_{β 1} 0) \cap_{i = 2}^{3} G_{F i}^{\|} (T_{a i} T_{b i} T_{c i}; R_{α i} R_{β i} R_{γ i})$	1.6,1.6

Table 6.

Typical compositions for the G_F Set $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

Compositions	Condition NO.
$\cap_{i = 1}^{4} G_{F i}^{\|} (T_{a i} T_{b i} 0; R_{α 1} R_{β i} R_{γ i})$	2.5,2.2,2.2
$G_{F i}^{\|} (T_{a 1} T_{b 1} 0; R_{α 1} R_{β i} 0) \cap_{i = 2}^{4} G_{F i}^{\|} (T_{a i} T_{b i} T_{c i}; R_{α i} R_{β i} 0)$	2.3,2.3,2.3
$G_{F i}^{\|} (T_{a 1} T_{b 1} 0; R_{α 1} R_{β i} 0) \cap_{i = 2}^{4} G_{F i}^{\|} (T_{a i} T_{b i} T_{c i}; R_{α i} R_{β i} R_{γ i})$	2.4,2.4,2.4

Table 7.

Typical compositions for the G_F Set $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c} \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

Compositions	Condition NO.
$\cap_{i = 1}^{5} G_{F i}^{\|} (T_{a i} T_{b i} T_{c i}; R_{α 1} R_{β i} 0)$	3.1,3.1,3.1,3.1
$G_{F 1}^{\|} (T_{a 1} T_{b 1} T_{c i}; R_{α 1} R_{β 1} 0) \cap_{i = 2}^{5} G_{F i}^{\|} (T_{a i} T_{b i} T_{c i}; T_{α i} R_{β i} R_{γ i})$	3.2,3.2,3.2,3.2

6. Design of Kinematic Limbs with Specific G_F Sets

6.1. Axes movement theorems

For designing kinematic limbs with specific G_F sets, we propose two theorems in terms of the axis movement. By using the two theorems, more kinematic limbs can be deduced from the simple kinematic limbs.

Theorem of axis movement in a 2-D plane. On one plane, if a kinematic chain has 2-D translations and one-dimensional rotation perpendicular to the 2-D translation plane, no matter where the revolute joint is the end-effector of the kinematic chain can rotate around any axis so as to be perpendicular to the 2-D plane on the end-effector.

Theorem of axes movement in 3-D space. In a 3-D space, if a kinematic chain has 3-D translations and rotations, no matter where the rotation joints are the end-effector of the kinematic chain can be around any point of the end-effector so as to rotate with the same rotations of the joints.

6.2. Design of kinematic limbs with specific characteristics

In order to synthesize the FG2RPM, we need to first design kinematic limbs with the required characteristics. According to the type of compositions listed in Table 5–7, the necessary kinematic limbs include those limbs with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ , $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0; & R_{α} & R_{β} & 0 \end{matrix})$ , $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0; & R_{α} & R_{β} & R_{γ} \end{matrix})$ , $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c}; & R_{α} & R_{β} & 0 \end{matrix})$ and $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c}; & R_{α} & R_{β} & R_{γ} \end{matrix})$ .

By using the 2-D or 3-D axis movement theorems, we can generate kinematic limbs with specific characteristics from the simple and intuitive kinematic limbs that contain traditional joints and composite joints [17]. Consequently, several typical kinematic limbs with the above-mentioned five different characteristics are sketched in Figure 1–4 and listed in Table 8, respectively. In particular, the composite joint $U^{*}$ is employed in Figure 4d and Figure 4e. It is important to point out that the kinematic limbs sketched in Figure 1–4 are only a small part of the possible limbs with specific characteristics and that more limbs can be designed according to the 2-D and 3-D axes movement theorems.

Figure 1.

Kinematic limbs with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ : (a) ${P(RR)}_{o}$ , (b) ${(CR)}_{o}$

Figure 2.

Kinematic limbs with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0; & R_{α} & R_{β} & 0 \end{matrix})$

Figure 3.

Kinematic limbs with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0; & R_{α} & R_{β} & R_{γ} \end{matrix})$

Figure 4.

Kinematic limbs with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c}; & R_{α} & R_{β} & 0 \end{matrix})$

Table 8.

Kinematic limbs with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c}; & R_{α} & R_{β} & R_{γ} \end{matrix})$

Joint	Limbs	Joint	Limbs
P , S	PPPS	U , P , S	UPS
	SPS	U , P , S	PUS
	PSS	U* , P , S	PU*S
R , S	SRS	U* , P , S	U*PS
R , S	RSS	P_a , S	P_aP_aP_aS

7. Type synthesis examples

A parallel mechanism normally consists of a moving platform which is connected to a fixed base by at least two kinematic limbs. By using the results listed in Table 5, Table 6 and Table 7, we can synthesize FG2RPM with the desired characteristics of directivity, provided that the corresponding requirements are fulfilled. That is, we could synthesize FG2RPM by combing kinematic limbs with the required characteristics under the corresponding conditions listed in Table 5, Table 6 and Table 7.

Figure 5 illustrates two parallel mechanisms with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ , from which we can see that the mechanism shown in Figure 5a has a symmetrical structure, whereas the mechanism shown in Figure 5b is non-symmetrical. Moreover, any two of the three rotation centres of the three kinematic limbs shown in Figure 5a coincide with each other. The translational plane of $l e g 1$ is parallel to the translational plane of $l e g 2$ , whereas the translational plane of $l e g 3$ is not parallel to the translational plane of $l e g 1$ or $l e g 2$ . Consequently, we can see that the parallel mechanism shown in Figure 5a is synthesized according to the first case listed in Table 5. Furthermore, the parallel mechanism shown in Figure 5b is synthesized via the second case listed in Table 5.

Figure 5.

Parallel mechanisms with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

In Figure 6, two symmetrical parallel mechanisms with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$ are sketched. The parallel mechanism shown in Figure 6a is synthesized according to the first case of Table 6. It is easy to see that the rotation centres of $l e g 3$ and $l e g 4$ coincide with each other. Moreover, the three points, including the coincided rotation centres, the rotation centre of $l e g 1$ and the rotation centre of $l e g 2$ , are collinear. All of the translational planes of the four limbs are parallel. Consequently, the parallel mechanism shown in Figure 6a features two-dimensional rotations, including the first one-dimensional rotation around the line passing the rotation centre of $l e g 1$ and the rotation centre of $l e g 2$ as well as the second one-dimensional rotation whose axis is perpendicular to the translational plane of any leg. Similarly, the parallel mechanism shown in Figure 6b is synthesized according to the first case in Table 6. In particular, the two-dimensional rotations are the first one-dimensional rotation around the line passing the two coincided rotation centres, and the second one-dimensional rotation whose axis is perpendicular to the translational plane of any leg.

According to the first case and the second case listed in Table 7 and the related kinematic limbs, we can synthesize two parallel mechanisms, shown in Figure 7, with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c}; & R_{α} & R_{β} & 0 \end{matrix})$ . From Figure 7a, we can see that all of the revolute joints connected to fixed base are parallel and that all of the revolute joints connected to the moving platform are parallel too. On the other hand, there is no special requirement for the construction of limbs for the parallel mechanism shown in Figure 7b.

Figure 6.

Parallel mechanisms with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0; & R_{α} & R_{β} & 0 \end{matrix})$

Figure 7.

Parallel mechanisms with the characteristics of $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c}; & R_{α} & R_{β} & 0 \end{matrix})$

8. Conclusion

This paper presents the type synthesis of parallel mechanisms - particularly the FG2RPM - according to the G_F sets theory. The main contributions of this work are as follows.

The concept of G_F sets can be employed to describe the characteristics of robotic manipulators.

The rule of two-dimensional rotations is utilized to lay the theoretical foundation for the intersection of G_F sets.

Typical compositions of characteristics are listed for the FG2RPM.

Two axes movement theorems are proposed for the design of kinematic limbs used for the synthesis of FG2RPM.

Several FG2RPM are synthesized and sketched to show the effectiveness of the proposed methodology.

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(2002). New kinematic structures for 2-, 3-, 4-, and 5-DOF parallel manipulator designs. Mech Mach Theory, 37(11): 1395–1411.

Type Synthesis of Parallel Mechanisms with the First Class G F Sets and Two-Dimensional Rotations

Abstract

Keywords

1. Introduction

2. Relevant results of GF sets theory

3. Rule of two-dimensional rotations

4. Intersection algorithms of GF sets

4.1. The Intersection Algorithms for the GFF Set G F I ( T a 0 0 ; R α R β 0 )

4.2. The Intersection Algorithms for the GFF Set G F I ( T a T b 0 ; R α R β 0 )

4.3. The Intersection Algorithms for the GFF Set G F I ( T a T b T c ; R α R β 0 )

5. Compositions of the characteristics of limbs for the FG2RPM

6. Design of Kinematic Limbs with Specific GF Sets

6.1. Axes movement theorems

6.2. Design of kinematic limbs with specific characteristics

7. Type synthesis examples

8. Conclusion

References

2. Relevant results of G_F sets theory

4. Intersection algorithms of G_F sets

4.1. The Intersection Algorithms for the G_FF Set $G_{F}^{I} (\begin{matrix} T_{a} & 0 & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

4.2. The Intersection Algorithms for the G_FF Set $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & 0 \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

4.3. The Intersection Algorithms for the G_FF Set $G_{F}^{I} (\begin{matrix} T_{a} & T_{b} & T_{c} \end{matrix}; \begin{matrix} R_{α} & R_{β} & 0 \end{matrix})$

6. Design of Kinematic Limbs with Specific G_F Sets