New Structural Representation and Digital-Analysis Platform for Symmetrical Parallel Mechanisms

Abstract

Abstract An automatic design platform capable of automatic structural analysis, structural synthesis and the application of parallel mechanisms will be a great aid in the conceptual design of mechanisms, though up to now such a platform has only existed as an idea. The work in this paper constitutes part of such a platform. Based on the screw theory and a new structural representation method proposed here which builds a one-to-one correspondence between the strings of representative characters and the kinematic structures of symmetrical parallel mechanisms (SPMs), this paper develops a fully-automatic approach for mobility (degree-of-freedom) analysis, and further establishes an automatic digital-analysis platform for SPMs. With this platform, users simply have to enter the strings of representative characters, and the kinematic structures of the SPMs will be generated and displayed automatically, and the mobility and its properties will also be analysed and displayed automatically. Typical examples are provided to show the effectiveness of the approach.

Keywords

Symmetrical Parallel Mechanism Structural Representation Mobility Analysis Digital-analysis Platform

1. Introduction

Symmetrical parallel mechanisms (SPMs) not only have the advantage of high rigidity, high strength, high load carrying capacity, low inertia, and good dynamic performance, but also possess the property of isotropy making them adaptable to various different tasks [1–2]. In the past decades, many scholars [3–8] have conducted research into SPMs using various approaches. With the development of computer technology, the methodology of digital research into mechanisms is a new area of research, which concerns how computers can carry out analyses, thus saving on human labour, and improving automation and intelligence in mechanical conceptual design [9]. Freudenstein [10] was the first to attempt the computer-based automatic sketching of mechanisms for conceptual design. Later, Mruthyunjaya [11], Yan [12] and Wang [13] promoted the development of a methodology for the creative synthesis of mechanisms. However, there is still no automatic digital-analysis platform for mobility analysis and kinematic structure sketching of SPMs.

The structural representation of SPMs is one of the most significant issues in this field of study. It should include all the topological information of SPMs so that these can be identified by the computer. The traditional structural representation of an SPM consists of the number of limbs and the types of kinematic pairs in a limb[14]. For example, 3-UPU denotes an SPM which possesses three limbs, each of which contains a universal pair, a prismatic pair and a universal pair from the base to the moving platform in turn. While the simplicity of this representation make it visually pleasing, it however results in the key information of the geometrical relationships between joint axes being left out, which affect such important kinematic characteristics of SPMs as mobility and its properties. Geometrical relationships between joint axes were first proposed by Dizioglu [15] in Germany. To describe these relationships, Huang and Li [2,16] proposed a representation using superscript and subscript symbols. Kong and Gosselin [17–19] proposed a concept, “leg-group”, and a form of representation using some special notations. No matter what the representation might be, if it is to be used in the automation of the conceptual design of SPMs, one property of the representation must be a one-to-one correspondence between the strings of representative characters and the kinematic structures of the SPMs. However, this is the very problem which has not yet been solved and is also one important reason why an automatic digital-analysis platform for mobility analysis and kinematic structure sketching of SPMs has not yet been developed.

The mobility of mechanisms is the first aspect to be considered in the study of kinematics and the dynamics of mechanisms [20]. Besides, what needs to be known is not only the mobility itself, but also the mobility's properties concerning the numbers of rotations and translations. A great many methods for mobility analysis have been put forward [22–26], whereby the work would be done manually. However, a digital-platform for automatic analysis has not been developed, the lack of which hampers the automation of the conceptual stage of mechanical design. Methods [2, 18, 21, 22, 26] based on the screw theory, the calculation of which mainly involves matrix operations, have the potential advantages of automatic analysis. However, there is no evidence as yet that the process of obtaining screws can be carried out by computers using the existing methods, which means automatic mobility analysis cannot be carried out.

In this paper, a new method of structural representation and a method for the programmable mobility analysis of SPMs are proposed. Based on these methods, a conceptual digital-analysis platform was created, using which the sketching of the kinematic structures of SPMs and the analysis of mobility and its properties can all be carried out automatically. Several examples are given to verify the effectiveness of the method.

2. Structural representation of SPMs

An SPM consists of a base (fixed platform), a moving platform and several identical structural limbs connecting the two platforms with geometrical symmetry. The traditional representation of SPMs is simple, but it does not reveal the basic geometrical relationships between joint axes which affect the kinematic characteristics of SPMs, such as mobility and its properties. Here, a new structural representation of SPMs is proposed, which consists of a structural representation of the platforms (the base and the moving platform) and a structural representation of the limbs.

2.1 Structural representation of the platforms

In an SPM, the two end joints of a limb are connected with the two platforms. The relationships between the axis of an end joint and a platform (line-plane relationships) can be classified into three basic types: parallel, vertically intersecting and generally intersecting, denoted by the characters ‘/’, ‘+’ and ‘∧’, respectively. A platform is usually connected to two or more limbs. The relationships between the axes of the end joints which are connected to a platform (relationships between rotational symmetrical lines) can also be classified into three types: parallel to one another, intersecting at one point, and forming a polygon, denoted by the characters ‘/’, ‘∧’, and ‘#’, respectively.

Based on the characteristics of the SPMs, the structural representation of a platform can take one of the following five types.

The axes of the end joints which are connected with the platform intersect vertically (+) with the platform. In this case, all these axes are also parallel (/). So the structural representation of the platform is ‘+/’, as shown in Fig.1(a).

The axes of the end joints connected with the platform generally intersect (∧) with the platform. In this case, these axes also intersect at one point (∧) in space. So the structural representation of the platform is ‘∧∧’, as shown in Fig.1(b).

If the axes of the end joints which are connected to the platform are parallel (/) to the platform, it takes one of the following types (3)–(5).

The axes are also parallel (/). The structural representation of the platform is ‘//’, as shown in Fig.1(c).

The axes intersect (∧). The structural representation of the platform is ‘/∧’, as shown in Fig.1(d).

The axes form a polygon (#). The structural representation of the platform is ‘/#’, as shown in Fig.1(e).

Figure 1.

Types of geometrical relationships on the platforms

So for SPMs, there are five types of platforms, as shown in Table 1.

Table 1.

Five types of platforms and their representation characters

Relationship between end joint axes and platform	Vertical	Intersecting	Parallel
Relationship between end joint axes	Parallel	Intersecting at one point	Parallel	Intersecting at one point	Constituting a polygon
Character representing the platform	+/	∧∧	//	/∧	/#

2.2 Structural representation of the limbs

Each limb in an SPM is a serial kinematic chain the structural representation of which consists of the types of joints and the geometrical relationships between the joint axes in the limb.

The kinematic pairs of SPMs are denoted by symbols: P for a prismatic pair, R for a revolute pair, C for a cylindrical pair, U for a universal pair, H for a helical pair and S for a spherical pair. As we know, a U pair can be regarded as a combination of two R pairs with intersecting and orthogonal axes, an S pair can be regarded as a combination of three R pairs with concurrent axes, and a C pair can be regarded as a combination of one R pair and one P pair with the same joint axis.

In a limb, the geometrical relationships between the joint axes fall into five types: parallel (‘/’), vertical and intersecting (‘+’), vertical but not intersecting (‘!’), intersecting (‘∧’), and skew lines (‘&’). If three joint axes in a limb are intersecting at one point, they are denoted as ‘*’. The geometrical relationships between joint axes in a limb and the characters representing them are listed in Table 2.

Table 2.

Representation characters between joint axes in a limb

Relationship	Parallel	Vertical but not intersecting	Vertical and intersecting	Intersecting	Skew lines	Three joint axes intersecting
Character	/	!	+	ˆ	&	^*

One of the limbs in an SPM can be chosen as the reference limb (RL), and we only need to know the representation of the RL in order to know those of the other limbs with the same structures. In the RL, the kinematic pairs are labelled in turn with the numbers 1˜j (j denotes the number of kinematic pairs in the RL) from the base to the moving platform. Pair 1 is connected to the base and defined as the basic pair (BP).

A reference pair (RP) is needed to represent a joint axis in space. Usually, joint (i-1) is chosen as the reference pair for joint i. However, if one of the joints 1˜(i-2) is parallel to, vertical to or intersects the axis of joint i, this joint instead of joint (i-1) is chosen as the reference pair. There is no need to choose a reference pair for the basic pair since its joint axis has been determined by the structural representation of the fixed platform. The representation of a pair i consists of the type of the pair, the label of the RP (omitted if joint (i-1) is the reference pair) and the geometrical relationship between the axes of the pair and its RP. In addition, if two or three joint axes in the RL intersect at a given point and the corresponding joint axes in the other limbs also intersect at the same point, character ‘|’ is added at the end to represent the case.

For example, the limb in Fig.2 has five R pairs, labelled with the numbers 1˜5. The axes of pairs 2 and 3 are parallel to that of pair 1, the axis of pair 4 intersects the axis of pair 3, and the axis of pair 5 is parallel to the axis of pair 4. Consequently, the RP of pairs 2 and 3 is pair 1, the RP of pair 4 is pair 3 and the RP of pair 5 is pair 4. Pair 1 is the BP and it is represented with ‘R’. The representations of pairs 2˜5 are ‘R/’, ‘R1/’, ‘R∧’ and ‘R/’, respectively. Therefore, the structural representation of the limb is ‘RR/R1/R∧R/’.

Figure 2.

A ‘RR/R1/R∧R/’ limb

In Fig.3, the limb has four R pairs and one P pair. The axis of pair 2 is perpendicular and intersects with pairs 1 and 3, the axes of pairs 3 and 4 are skew lines, the axis of pair 5 intersects the axis of pair 4, the axis of pair 3 is parallel to the axis of pair 1. So the representations of pairs 1˜5 are ‘R’, ‘P+’, ‘R1/’, ‘R&’ and ‘R∧’, respectively. Therefore, the structural representation of the limb is ‘RP+R1/R&R∧’.

Figure 3.

A ‘RP+R1/R&R∧’ limb

2.3 Structural representation of SPMs

The string of representative characters is used to represent the structure of a complete SPM. Its form is “representation of the base - representation of limbs - representation of the moving platform”.

In Fig.4(a), the SPM has three limbs. The three joint axes connected to the base are parallel to the base, and they form a polygon. So, the structural representation of the base is ‘3/#’. The structural representation of the moving platform is ‘3/#’. The reference limb consists of an R pair, a P pair and a C pair. The axis of the P pair vertically intersects with that of the R pair, and the axis of the C pair is parallel to that of the R pair. So the representation of the reference limb is ‘RP+C1/’. Therefore, the string of representative characters of the SPM is ‘3/#-RP+C1/-3/#’. Similarly, the strings of representative characters of the SPMs shown in Fig.4(b) and 4(c) are ‘3+/-RR/R+R∧R^*|-3∧∧’ and ‘3/#-RP+R1/R!R∧|-3∧∧’.

Figure 4.

Several SPMs

The string of representative characters of an SPM contains the geometrical relationships between the joint axes, so it is useful in the automatic sketching of kinematic structures and the automatic analysis of the mobility and its properties of the SPM.

3 Method for the automatic analysis of the mobility and its properties

In the screw theory[14, 24], a unit screw $ is defined by a pair of vectors:

$ = (S; S_{0}) = (S; r \times S + p S) = (a b c; d e f)

(1)

where S is a unit vector specifying the direction of the screw axis, r is the position vector of any point on the screw axis in a reference coordinate system, and p is called the pitch. ( S; S ₀) has six components, called the PlÜcker coordinates. Unit screws associated with a prismatic pair or a couple can be expressed as (0; S ) and unit screws associated with a revolute pair or a force are expressed as ( S; r × S ). A screw, $ ^r = ( S ^r; $ ^r₀), and a set of screws, $ ₁, $ ₂, … $ _j are said to be reciprocal if they satisfy the condition

$_{j} \circ $^{r} = S_{j} \cdot S_{0}^{​^{r}} + S_{0 j} \cdot S^{r} = 0

(2)

where “o” denotes the reciprocal product and $ _j is the j-th screw of the screw set. If $ _j denotes a twist, $ ^r denotes a wrench. Otherwise, if $ ^r denotes a twist, $ _j denotes a wrench. Eq.(2) can be also expressed as

$_{j} \circ $^{r} = {[Π $_{j}]}^{T} $^{r} = 0

(3)

where

Π = (\begin{matrix} 0 & I_{3} \\ I_{3} & 0 \end{matrix})

(4)

where I₃ is the 3×3 identity matrix and 0 is the 3×3 zero matrix.

Although the screw theory is good for mobility analysis, it cannot be directly applied to automatic mobility analysis [27]. Here, based on the string of representative characters and the screw theory, a fully-automatic method for the analysis of the mobility and its properties is presented.

Step 1: Establishing a coordinate system

The establishment of a coordinate system is the basis for obtaining screws. For a given SPM, any one of its limbs can be chosen as the RL. In general, we label the limbs with the numbers 1, 2, 3˜k, and limb 1 is chosen as the RL. The coordinate system O-XYZ is determined by the following rules: if two or more joint axes intersect at one point in the RL and the corresponding joint axes in all the other limbs also intersect at the same point, the intersecting point is used as the origin. The X-axis is parallel to the line connecting the centre point of the base and the centre point of the BP of the RL. The Z-axis is perpendicular to the base plane and goes upward. The Y-axis is obtained using the right-hand rule. Here, this is called mode 1. Otherwise, if there is no common point of intersection, the coordinate origin is the centre point of the base plane. The X-axis is the line connecting the origin coordinate with the centre point of the BP of the RL. The Z-axis is also perpendicular to the base plane and goes upward. The Y-axis is obtained using the right-hand rule. Here, this is called mode 2.

Step 2: Obtain the twist screw system and the wrench screw system of the RL

If a pair is a P joint, only the joint axis needs to be known and its twist screw is expressed as (0; S ). For an R pair, generally, r is (l m n). Alternatively, if the joint axis is in the XOY coordinate plane, r is (l m 0); if it is in the YOZ coordinate plane, it is (0 m n); if it is in the XOZ coordinate plane, it is (l 0 n). However, if a joint axis passes through the origin of the coordinate system O-XYZ, r is (0 0 0). Based on the above coordinate system O-XYZ and the representation of the base, the twist screws of the BP in the RL can be obtained easily. Here, the twist screws for the R pair are given in Table 3, in which a1, b1 and c₁ denote the angles between the unit vector specifying the direction of the joint axis and the X-axis, Y-axis and Z-axis, respectively. For example, for mode 2 of the coordinate system, if the BP is an R pair and the representation of the base is ‘/∧’, the axis of the BP passes through the origin, and the axis and the X-axis are collinear. So, S is (1 0 0), and r is (0 0 0). The twist screw of the BP is (1 0 0; 0 0 0).

Table 3.

Twist screws of the BP

Representation		/∧	/#	//	+/	∧∧
Mode 1	S	(1 0 0)	(0 1 0)	(1 0 0) or (0 1 0)	(0 0 1)	(cos(a₁) cos(b₁) cos(c₁))
	r	(0 0 0)	(l m 0)	(0 0 0) or (l m 0)	(l 0 n)	(l m n)
	$	( S; S×r )	( S; S×r )	( S; S×r )	( S; S×r )	( S; S×r )
Mode 2	S	(1 0 0)	(0 1 0)	(1 0 0) or (0 1 0)	(0 0 1)	(cos(a₁) cos(b₁) cos(c₁))
	r	(l m n)	(l m n)	(l m n)	(l m n)	(l m n)
	$	( S; S×r )	( S; S×r )	( S; S×r )	( S; S×r )	( S; S×r )

The twist screws of the other pairs in the RL can be obtained from the geometrical relationships and the twist screws of the RPs. The axis direction of a RP may be parallel to the X-axis, namely, S ₁= (1 0 0), and r ₁= (l₁ m₁ n₁), and all the types of twist screws of the other pairs in different geometrical relations are listed in Table 4. Similarly, twist screws can be obtained if the axis direction of the RP is parallel to the Y-axis or the Z-axis. Table 5 lists the twist screws of all kinds of other pairs in different geometrical relations when the axis direction of the RP is parallel to O-YZ plane, namely, S1= (0 cos(b 1) cos(c1)) and r ₁= (l₁ m₁ n₁). Similarly, the twist screws of other pairs can also be obtained if the axis direction of the RP is parallel to the O-XY plane or the O-XZ plane. Table 6 lists the all the types of twist screws of the other pairs in different invariant geometrical relations when the RP is situated in a general space, namely, S ₁ = (cos(α₁), cos(β₁), cos(γ₁)) and r ₁ = (l₁, m₁, n₁). In Tables 4˜6, a_i, b_i and c_i denote the angles between the unit vectors and the X-, Y- and Z-axis, respectively.

Table 4.

Twist screws of other pairs for the RP (S₁= (0 cos(b₁) cos(c₁)), r₁= (l₁ m₁ n₁))

	S	r	$
Twist screw of the RP	(1 0 0)	(l₁ m₁ n₁)	( S; S×r )
Parallel(/)	(1 0 0)	(l₂ m₂ n₂)	( S; S×r )
Vertical and intersecting(+)	(0 cos(b₃) cos(c₃))	(l₁ m₁ n₁)	( S; S×r )
Vertical (!)	(0 cos(b₄) cos(c₄))	(l₄ m₄ n₄)	( S; S×r )
Intersection (∧)	(cos(a₅) cos(b₅) cos(c₅))	(l₁ m₁ n₁)	( S; S×r )
skew(&)	(cos(a₆) cos(b₆) cos(c₆))	(l₆ m₆ n₆)	( S; S×r )

Table 5.

Twist screws of other pairs for the RP (S₁= (1 0 0), r₁= (l₁ m₁ n₁))

	S	r	$
Twist screw of the RP	(0 cos(b₁) cos(c₁))	(l₁ m₁ n₁)	( S; S×r )
Parallel(/)	(0 cos(b₁) cos(c₁))	(l₂ m₂ n₂)	( S; S×r )
Vertical and intersecting(+)	(cos(a₃) cos(b₃) -cos(b₃)cos(b₁)/cos(c₁))	(l₁ m₁ n₁)	( S; S×r )
Vertical (!)	(cos(a₄) cos(b₄) -cos(b₄)cos(b₁)/cos(c₁))	(l₄ m₄ n₄)	( S; S×r )
Intersection (∧)	(cos(a₅) cos(b₅) cos(c₅))	(l₁ m₁ n₁)	( S; S×r )
skew(&)	(cos(a₆) cos(b₆) cos(c₆))	(l₆ m₆ n₆)	( S; S×r )

Table 6.

Twist screws of other pairs for the RP (S₁= (cos(a₁) cos(b₁) cos(c₁)), r₁= (l₁ m₁ n₁))

	S	r	$
Twist screw of the RP	(cos(a₁) cos(b₁) cos(c₁))	(l₁ m₁ n₁)	( S; S×r )
Parallel(/)	(cos(a₁) cos(b₁) cos(c₁))	(l₂ m₂ n₂)	( S; S×r )
Vertical and intersecting(+)	(cos(a₃) cos(b₃) –(cos(a₃)cos(a₁) +cos(b₃)cos(b₁))/cos(c₁))	(l₁ m₁ n₁)	( S; S×r )
Vertical (!)	(cos(a₄) cos(b₄) –(cos(a₄)cos(a₁) +cos(b₄)cos(b₁))/cos(c₁))	(l₄ m₄ n₄)	( S; S×r )
(cos(a₅) cos(b₅) cos(c₅))	(l₁ m₁ n₁)	( S; S×r )
skew(&)	(cos(a₆) cos(b₆) cos(c₆))	(l₆ m₆ n₆)	( S; S×r )

Taking the string of representative characters ‘3/#-RP+C1/-3/#’ for example, the origin of the coordinate system O-XYZ is the centre point of the base plane. Thus, the twist screw of the BP is (0 1 0; (l m 0)×(0 1 0)), in which S ₁ is (0 1 0) and r is (l m 0), according to Table 3. ‘P+’ denotes that the axis of the P pair is vertical with that of the BP, so S ₂ is (cos(a₂) 0 cos(c₂)). ‘C1/’ denotes that the axis of the P pair is parallel to that of the BP, so S ₃ is (0 1 0). Consequently, the twist screw system of the RL is $ ₁₁ = (0 1 0; (l m 0)×(0 1 0)), $ ₁₂ = (0 0 0; cos(a₂) 0 cos(c₂)), $ ₁₃ = (0 1 0; (l₃ m₃ n₃)×(0 1 0)), $ ₁₄ = (0 0 0; 0 1 0). Note that the C pair is expressed with one R pair and one P pair with the same axis.

With this method, the twist screw system, $ ₁, of the RL can be obtained as

$_{1} = {({($_{11})}^{T}, {($_{12})}^{T}, {($_{13})}^{T}, \dots, {($_{1 j})}^{T})}^{T}

(5)

where j denotes the number of twist screws.

According to the reciprocal product $ ₁ o $ ₁^r=0, the wrench screw system $ ₁^r of the RL can be solved as

$_{1}^{r} = {({($_{11}^{r})}^{T}, {($_{12}^{r})}^{T}, {($_{13}^{r})}^{T}, \dots, {($_{1 h}^{r})}^{T})}^{T}

(6)

where, h=6-j, h is the number of the wrench screws of the RL.

Step 3: Solving the wrench screw systems of the other limbs

The wrench screw system, $ ₁^r, of the ith limb can be obtained by rotating the wrench screw system $ ₁^r by θ_i around the Z-axis

$_{i}^{r} = T (θ_{i}) $_{1}^{r}

(7)

The limb number of an SPM is denoted as k, thus,

θ_{i} = \frac{2 π}{k} (i - 1)

(8)

Then

T (θ_{i}) = [\begin{matrix} R (θ_{i}) & 0 \\ 0 & R (θ_{i}) \end{matrix}]

(9)

where

R (θ_{i}) = [\begin{matrix} c o n θ_{i} & - \sin θ_{i} & 0 \\ \sin θ_{i} & c o n θ_{i} & 0 \\ 0 & 0 & 1 \end{matrix}]

(10)

Consequently, the wrench screw systems, $ ^r, of all the limbs can be solved, i.e.,

$^{r} = {({($_{1}^{r})}^{T}, {($_{2}^{r})}^{T}, \dots, ($_{k}^{r})^{T})}^{T}

(11)

Step 4: Solving the maximum linear independent group (denoted by $ _m^r here) of the wrench screw systems $ ^r. Then $ _m^r represents the constraint of the moving platform exerted by all the limbs of the mechanism. The rank, D, of the wrench screw set which consists of the wrench screw systems $ ^r can be obtained as

D = r a n k ($^{r}) = r a n k ($_{m}^{r})

(12)

Step 5: Computing the mobility and analysing its properties. The mobility of the moving platform (denoted by F here) can be computed by

F = 6 - D

(13)

Mobility properties can be analysed by solving the reciprocal screw (denoted by $ _m here) of $ _m^r based on $ _m o $ _m^r=0. In fact, $ _m is simply the twist screw of the moving platform, which includes the mobility properties of the moving platform.

Example 1 : The string of representative characters for the mechanism in Fig.5(a) is ‘4/#-RR/R1/-4/#’. The mobility and its properties can be analysed as follows.

Figure 5.

Two SPMs

Step 1: The central point of the base is chosen as the origin of the coordinate system according to the rules (mode 2).

Step 2: The representation of the base is ‘4/#’ and the twist screw of the BP can be expressed as $ ₁₁ = (0 1 0; (l m 0)×(0 1 0)). The representation of the RL is ‘RR/R1/’, so, $ ₁₂ = (0 1 0; (l₂ m₂ n₂)×(0 1 0)), and $ ₁₃ = (0 1 0; (l₃ m₃ n₃)×(0 1 0)). The twist screw system of the RL(limb 1) is

$_{1} = (\begin{array}{l} 0 1 0; 0 0 1 \\ 0 1 0; - n_{2} 0 l_{2} \\ 0 1 0; - n_{3} 0 l_{3} \end{array})

(14)

According to the reciprocal product relationship, the wrench screw system is

$_{1}^{r} = (\begin{array}{l} 0 1 0; 0 0 0 \\ 0 0 0; 0 0 1 \\ 0 0 0; 1 0 0 \end{array})

(15)

Step 3: The other three wrench screw systems $ ₂^r and $ ₃^r can be achieved by rotating $ ₁^r by π/2, π and 3π/2 around the Z-axis, respectively.

\begin{array}{c} $_{2}^{r} = (\begin{array}{l} - 1 0 0; 0 0 0 \\ 0 0 0; 0 0 1 \\ 0 0 0; 0 1 0 \end{array}), $_{3}^{r} = (\begin{array}{l} 0 - 1 0; 0 0 0 \\ 0 0 0; 0 0 1 \\ 0 0 0; - 1 0 0 \end{array}), \\ $_{4}^{r} = (\begin{array}{l} 1 0 0; 0 0 0 \\ 0 0 0; 0 0 1 \\ 0 0 0; 0 - 1 0 \end{array}) \end{array}

(16)

Step 4: The maximum linear independent group, $ _m^r, can be obtained as

$_{m}^{r} = (\begin{array}{l} 10 0; 0 0 0 \\ 0 - 1 0; 0 0 0 \\ 0 0 0; - 1 0 0 \\ 0 0 0; 0 1 0 \\ 0 0 0; 0 0 1 \end{array})

(17)

The rank, D, of the wrench screw set which consists of the wrench screw systems, $ ^r, can be obtained as

D = r a n k ($_{m}^{r}) = 5

(18)

Step 5: The mobility of the moving platform is F=6-D=1. The twist screw of the moving platform is $ _m =[0 0 0; 0 0 1]. Consequently, the mobility property of the mechanism is a translation along the Z-axis.

Example 2 : An SPM, the string of representative characters of which is ‘3/#-RP+R1/R!R∧|-3∧∧’, is shown in Fig.5(b).

Step 1: Some corresponding joint axes in the limbs intersect at one point. This point is used as the origin of the coordinate system (mode 1). The X-axis is parallel to the line connecting the origin coordinate with the central point of the BP of the RL. The Z-axis is perpendicular to the base plane and goes upward.

Step 2: The representation of the base is ‘3/#’, so, $ ₁₁ = (0 1 0; (l m n)×(0 1 0)). The representation of the RL is ‘RP+R1/R!R∧|’, so, $ ₁₂ = (0 0 0; cos(a₂) 0 cos(c₂)), $ ₁₃ = (0 1 0; (l₃ m₃ n₃)×(0 1 0)), $ ₁₄ = (cos(a₄) 0 cos(c₄); 0 0 0), and $ ₁₅ = (cos(a₅) cos(b₅) cos(c₅); 0 0 0).

The twist screw system $ ₁ of the RL (limb 1) is

$_{1} = (\begin{array}{l} 0 1 0;  - n 0 l \\ 0 0 0; c o s (a_{2}) 0 c o s (c_{2}) \\ 0 1 0; - n_{3} 0 l_{3} \\ c o s (a_{4}) 0 c o s (c_{4}); 0 0 0 \\ c o s (a_{5}) c o s (b_{5}) cos(c_{5}); 0 0 0 \end{array})

(19)

So, the wrench screw system of the RL is

$_{1}^{r} = (0 1 0; 0 0 0)

(20)

Step 3: The other two wrench screw systems, $ ₂^r and $ ₃^r, can be obtained by rotating $ ₁^r by 2π/3 and 4π/3 around the Z-axis direction respectively.

\begin{array}{l} $_{2}^{r} = (- \sqrt{3} / 2, - 1 / 2,0; 0, 0, 0), \\ $_{3}^{r} = (\sqrt{3} / 2, - 1 / 2, 0; 0, 0, 0) \end{array}

(21)

Step 4: The maximum linear independent group, $ _m^r, can be obtained as

$_{m}^{r} = (\begin{array}{l} 0, 1, 0; 0, 0, 0 \\ - \sqrt{3} / 2, - 1 / 2, 0; 0, 0, 0, \end{array})

(22)

The rank, D, of the wrench screw set which consists of the wrench screw systems $ ^r can be obtained as

D = r a n k ($_{m}^{r}) = 2

(23)

Step 5: The mobility of the moving platform is F=6-D=4. The twist screw of the moving platform is

$_{m} = (\begin{array}{l} 0 0 1; 0 0 0 \\ 0 0 0; 1 0 0 \\ 0 0 0; 0 1 0 \\ 0 0 0; 0 0 1 \end{array})

(24)

which denotes that the moving platform has three rotational freedoms around the X-axis, Y-axis and Z-axis, respectively and a translation freedom along the Z-axis.

4. Digital-analysis platform

Following the approach above, a digital-analysis platform for SPMs was developed successfully for the first time. Using this platform the kinematic structures of SPMs can be sketched and the mobility of SPMs can be analysed automatically based on the input of the structure in the form of a string of representative characters. The flow chart of the digital-analysis platform is shown in Fig.6. For the sake of convenience, when sketching the kinematic structures of mechanisms, some symbols are used to represent different pairs: for the R pair, for the P pair, for the H pair, for the U pair, for the C pair, and for the S pair, as shown in Table 6.

Figure 6.

The flow chart of the digital-analysis platform

Table 6.

Kinematic pairs and their corresponding symbols

Type of pairs	R pair	P pair	H pair	U pair	C pair	S pair
Symbols

5. Several typical examples

In the twist screws of the limbs or the moving platform, if one component of S or S ₀ is not 0 or 1, the component can be represented by a letter, such as a, b, or c, etc., to simplify the representation of the types of screws. This does not affect the results.

Example 3: Input the representations of the limb ‘RP+C1/’, the base ‘3/’, and the moving platform ‘3/#’. Then the kinematic structure of the SPM is sketched automatically by the software platform, as shown in Fig.7. Click the button “Mobility”, and the coordinate system in the kinematic structure is established with the automatically selected RL. The mobility analysis is carried out automatically. Click the button “Twist screw of limb”, and the twist screw system of the RL is obtained, as shown in Fig.8. Click the button “Twist screw of platform”, and the twist screw of the moving platform is obtained, as shown in Fig.9. According to Fig.7, the platform of the SPM has three translational degrees of freedom along the X-axis, Y-axis and Z-axis.

Figure 7.

The analysis of the ‘3/#-RP+C1/-3/#’ SPM based on the developed platform

Figure 8.

Twist screw of the RL

Figure 9.

Twist screw of the moving platform

Example 4: Input the representations of the limb ‘RR/R1/’, the base ‘4/#’, and the moving platform ‘4/#’. The software automatically sketches the kinematic structure of the SPM and analyses its mobility. As shown in Fig.10, the moving platform of the SPM has one translational freedom along the Z-axis. The twist screw system of the RL is shown in Fig.11 and the twist screw of the moving platform is shown in Fig.12. A comparison of the results with those from Example 1 shows the effectiveness of the platform.

Figure 10.

The analysis of the ‘4/#-RR/R-4/#’ SPM based on the developed platform

Figure 11.

Twist screw of the RL

Figure 12.

Twist screw of the moving platform

Example 5: Input the representations of the limb ‘RR/R/R!R∧|’, the fixed platform ‘3/#’ and the moving platform‘3∧∧’. The software automatically generates the kinematic structure and the mobility of the SPM. The twist screw system of the RL is shown in Fig.14 and the twist screw of the moving platform is shown in Fig.15. According to Fig.13, the moving platform of the SPM has four degrees of freedom which are three rotations and one translation along the Z-axis. A comparison of the results with those of Example 2 shows the effectiveness of the platform.

Figure 13.

The analysis of the ‘3/#-RP+R1/R!R∧|-3∧∧’ SPM based on the developed platform

Figure 14.

Twist screw of the RL

Figure 15.

Twist screw of the moving platform

6. Conclusions

In this paper, a new structural representation for SPMs is presented, with which strings of representative characters have a one-to-one correspondence with the structures of SPMs. The string of representative characters is intuitive and vivid enough for users to write and read easily. It is also identifiable by computers.

Based on the string of representative characters and the screw theory, a new method for analysing the mobility of SPMs is proposed, which is also legible for computers.

A conceptual digital-analysis platform was created for the first time, using which the kinematic structure of SPMs can be sketched and their mobility can be analysed automatically, based on a given string of representative characters. Several typical examples are given at the end of the paper, illustrating the effectiveness of the method. In brief, the digital-analysis platform developed here is effective and efficient in analysing the mobility of SPMs. It achieves the automation and intelligence of mobility analysis and structure sketching, which will lay a good foundation for the automation of mechanical conceptual design.

7. Acknowledgements

The authors are grateful to the project (No.51275437, 50905155) supported by NSFC, the Hebei Nature Science Foundation (No.E2012203154), the 2012 open project of State Key Laboratory of Mechanical Transmission (No. SKLMT-KFKT-201208) and the Programe of Excellent Young Scientists of the Higher colleges of Hebei province (No.CPRC017).

References

Huang

Q.C.

(2002) Type synthesis principle of minor-mobility parallel manipulators. Science in China, Series E: Technological Sciences 45(3): 241–248.

Huang

Q. C.

(2003) Type synthesis of symmetrical lower-mobility parallel mechanisms using the constraint-synthesis method. International Journal of Robotics Research 22(1): 59–79.

Liu

X. J.

Wang

J. S.

Pritschow

(2006) Kinematics, singularity and workspace of planar 5R symmetrical parallel mechanisms. Mechanism and Machine Theory 41(2):145–169.

Bonev

I. A.

Gosselin

C. M.

(2006) Analytical determination of the workspace of symmetrical spherical parallel mechanisms. IEEE Transactions on Robotics 22(5): 1011–1017.

Fang

H. R.

Gao

Y. L.

Fang

Y. F.

Liu

P. X.

(2008) Structure synthesis of symmetrical low-DOF parallel manipulators. 2007 IEEE International Conference on Robotics and Biomimetics:1948–1953.

Zhu

S. J.

Huang

Zhao

M. Y.

(2009) Singularity analysis for six practicable 5-DoF fully-symmetrical parallel manipulators. Mechanism and Machine Theory 44(4):710–725.

Goldsmith

P. B.

(2002) Kinematics and stiffness of a symmetrical 3-UPU translational parallel manipulator. Proceedings - IEEE International Conference on Robotics and Automation 4: 4102–4107.

Q. C.

Huang

(2003) A family of symmetrical lower-mobility parallel mechanisms with spherical and parallel subchains. Journal of Robotic Systems 20(6): 297–305.

Ding

H. F.

Huang

(2007) A new theory for the topological structure analysis of kinematic chains and its applications, Mechanism and Machine Theory 42(10):1264–1279.

10.

Freudenstein

Dobrjanskyj

(1964) On a theory of type synthesis of mechanisms. Proceedings - 11th International Congress of Applied Mechanics: 420–428

11.

Mruthyunjaya

T. S.

Raghavan

M. R.

(1984) Computer-Aided Analysis of the Structure of Kinematic Chains. Mechanism and Machine Theory 19(3): 357–368.

12.

Yan

H. S.

Hwang

Y. W.

(1990) Number synthesis of robots based on permutation groups. Mathematical and Computer Modeling 13 (8): 29–42.

13.

Wang

Y. X.

Hong

H. S.

(2002) Computerized rules-based regeneration method for conceptual design of mechanisms. Mechanism and Machine Theory 37(9): 833–849.

14.

Huang

Kong

L. F.

Fang

Y. F.

(1997) Mechanism theory of parallel robotic manipulators and control. Beijing: China Machine Press.

15.

Dizioglu

(1978) Theory and practice of spatial mechanisms with special positions of the axes. Mechanism and Machine Theory13(2):139–153.

16.

Huang

Q. C.

(2002) General Methodology for Type Synthesis of Symmetrical Lower-Mobility Parallel Manipulators and Several Novel Manipulators. The International Journal of Robotics Research 21(2):131–145.

17.

Kong

X. W.

Gosselin

C. M.

(2004) Type synthesis of 3-DOF PPR parallel manipulators based on screw theory and the concept of virtual chain. Proceedings of the ASME Design Engineering Technical Conference 2B:1251–1260

18.

Kong

X. W.

Gosselin

C.M.

(2005) Mobility analysis of parallel mechanisms based on screw theory and the concept of equivalent serial kinematic chain. Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference-DETC 7B: 911–920.

19.

Kong

X. W.

Gosselin

C. M.

(2004) Type synthesis of 3T1R 4-DOF parallel manipulators based on screw theory. IEEE Transactions on Robotics and Automation 20(2): 181–190.

20.

Huang

Liu

J. F.

Zeng

D. X.

(2009) A general methodology for mobility analysis of mechanisms based on constraint screw theory. Science in China, Series E: Technological Sciences 52(5): 1337–1347.

21.

Huang

Zhao

Y. S.

Zhao

T. S.

(2006) Advanced Spatial Mechanisms. Beijing: Higher Education Press.

22.

Zhao

J. S.

Zhou

(2004) A theory of degrees of freedom for mechanisms. Mechanism and Machine Theory 39(6): 621–643.

23.

Gogu

(2005) Mobility of mechanisms: A critical review. Mechanism and Machine Theory 40(9): 1068–1097.

24.

Zhao

J. S.

Feng

Z. J.

Dong

J. X.

(2006) Computation of the configuration degree of freedom of a spatial parallel mechanism by using reciprocal screw theory. Mechanism and Machine Theory 41(12): 1486–1504.

25.

Yang

D. C.

Xiong

Yang

X. D.

(2008) A simple method to calculate mobility with Jacobian. Mechanism and Machine Theory 43(9): 1175–1185.

26.

Dai

J. S.

Zhen

Harvey

(2006) Mobility of Overconstrained Parallel Mechanisms. Journal of Mechanical Design, Transactions of the ASME 128(1):220–229.

27.

Huang

Q. C.

Ding

H.F.

(2012) Theory of Parallel Mechanisms. Dordrecht: Springer.