Position-Singularity Analysis of a Class of the 3/6-Gough-Stewart Manipulators Based on Singularity-Equivalent-Mechanism

Abstract

This paper addresses the problem of identifying the property of the singularity loci of a class of 3/6-Gough-Stewart manipulators for general orientations in which the moving platform is an equilateral triangle and the base is a semiregular hexagon. After constructing the Jacobian matrix of this class of 3/6-Gough-Stewart manipulators according to the screw theory, a cubic polynomial expression in the moving platform position parameters that represents the position-singularity locus of the manipulator in a three-dimensional space is derived. Graphical representations of the position-singularity locus for different orientations are given so as to demonstrate the results. Based on the singularity kinematics principle, a novel method referred to as ‘singularity-equivalent-mechanism' is proposed, by which the complicated singularity analysis of the parallel manipulator is transformed into a simpler direct position analysis of the planar singularity-equivalent-mechanism. The property of the position-singularity locus of this class of parallel manipulators for general orientations in the principal-section, where the moving platform lies, is identified. It shows that the position-singularity loci of this class of 3/6-Gough-Stewart manipulators for general orientations in parallel principal-sections are all quadratic expressions, including a parabola, four pairs of intersecting lines and infinite hyperbolas. Finally, the properties of the position-singularity loci of this class of 3/6-Gough-Stewart parallel manipulators in a three-dimensional space for all orientations are presented.

1. Introduction

The past few decades, parallel manipulator systems have become one of the main focuses of attention in research in robotics. This popularity has been motivated by the fact that parallel manipulators possess certain specific advantages over serial manipulators, namely higher rigidity and load-carrying capacity, better dynamic performance and a simple inverse position kinematics, etc. Among them, the best-known parallel manipulator is the Gough-Stewart platform, which was introduced as an aircraft simulator by Stewart [1] in 1965.

One of the important problems in robot kinematics is their special configuration. As with parallel manipulators, in such configurations the moving platform gains at least one undesired degree of freedom (DOF) while all of the actuators are locked. Meanwhile, infinite active forces must be applied to balance the loads exerted on the moving platform, otherwise risking the mechanism to breakdown. Hence, the prediction of the singular configurations of a kinematic chain beforehand is a very challenging yet open topic in robotics kinematics.

Figure 1.

Schematic of a 3/6-Gough-Stewart parallel manipulator

This phenomenon has attracted much attention from many researchers. Hunt [2] first discovered a special configuration for the parallel manipulator that occurs when all of the segments associated with the prismatic actuators intersect a common line. Fichter [3] pointed out that a special configuration is attained when the moving platform rotates around a Z-axis by ψ=±π/2, whatever the position of the moving platform may be. Using the Grassman line geometry, Merlet [4-6] studied the singularity of the 3/6-Gough-Stewart parallel manipulators systematically. Gosselin and Angeles [7] showed that singularities of parallel manipulators could be classified into three different types based on the determinants of a parallel manipulator's Jacobian matrices. Recently, Mayer-St-Onge and Gosselin [8] pointed out that the position-singularity loci of a general Gough-Stewart manipulator should be a polynomial expression of degree three, and so on [9-16]. Most of the reported work in this area has been performed in geometric conditions of singular configurations, deriving the analytical expression and obtaining graphical representations of the singularity locus. However, to the best of the authors' knowledge, much less work on the singularity has been reported on the topic of the property identification of the singularity loci themselves, which is of great importance for the further understanding of the singularity, the most relevant investigations having been made by Huang et al. [13-15] for the property identification of the singularity loci of the 3/6-Gough-Stewart manipulator for special orientations when φ=±30°, ±90°, or ±150°, and ψ≠±30°, ±90°, ±150° and θ≠0°.

This paper will further analyse the property of the position-singularity loci of the 3/6-Gough-Stewart manipulator, as shown in Fig. 1, for general orientations. Its moving platform, B₁B₃B₅, is an equilateral triangle, and the base, C₁C₂, …, C₅C₆, is a semiregular hexagon. Our study is as follows. After constructing the Jacobian matrix of this class of 3/6-Gough-Stewart manipulators, an analytical expression that represents the position-singularity locus of the manipulator in a three-dimensional space for a constant orientation is derived. Graphical representations of the position-singularity locus of the manipulator for different orientations are illustrated with examples. Further, using the method of the singularity-equivalent-mechanism, a quadratic polynomial expression representing the position-singularity locus of the manipulator in the principal-section is derived, and the property of the position-singularity loci in parallel principal-sections is identified. Their geometric and kinematic properties are also researched in detail in the present paper.

2. The Jacobian matrix and moving reciprocal screws of the parallel manipulator at singular configurations

The Jacobian matrix of this class of 3/6-Gough-Stewart manipulators can be constructed, as follows, according to the theory of static equilibrium proposed in [11]:

$\begin{array}{l}{\left[J\right]}^T=\left[\begin{array}{cccc}{\$}_1& {\$}_2& {\$}_3{\$}_4& {\$}_5{\$}_6\end{array}\right]=\left(\begin{array}{cccccc}{S}_1& S_2& S_3& S_4& S_5& S_6\\ S_{O1}& S_{O\text{2}}& S_{O\text{3}}& S_{O\text{4}}& S_{O\text{5}}& S_{O\text{6}}\end{array}\right)\\ =\left(\begin{array}{cccccc}\frac{\text{(}{B}_1-C_1\text{)}}{\Vert B_1-C_1\Vert }& \frac{\text{(}{B}_1-C_2\text{)}}{\Vert B_1-C_2\Vert }& \frac{\text{(}{B}_3-C_3\text{)}}{\Vert B_3-C_3\Vert }& \frac{\text{(}{B}_3-C_4\text{)}}{\Vert B_3-C_4\Vert }& \frac{\text{(}{B}_5-C_5\text{)}}{\Vert B_5-C_5\Vert }& \frac{\text{(}{B}_5-C_6\text{)}}{\Vert B_5-C_6\Vert }\\ \frac{\text{(}{C}_1\times B_1\text{)}}{\Vert B_1-C_1\Vert }& \frac{\text{(}{C}_2\times B_1\text{)}}{\Vert B_1-C_2\Vert }& \frac{\text{(}{C}_3\times B_3\text{)}}{\Vert B_3-C_3\Vert }& \frac{\text{(}{C}_4\times B_3\text{)}}{\Vert B_3-C_4\Vert }& \frac{\text{(}{C}_5\times B_5\text{)}}{\Vert B_5-C_5\Vert }& \frac{\text{(}{C}_6\times B_5\text{)}}{\Vert B_5-C_6\Vert }\end{array}\right)\end{array}$ (1)

where vectors, B _i , C _j (i=1, 3, 5, j=1, 2, …, 5, 6), respectively denote the vertex vectors of the moving platform and the base with respect to the fixed reference frame defined in Section 3; $ _i (i=1, 2, …, 6) is a line vector connecting two vertices B _i , C _j of the moving and base platforms. Its Plücker coordinates are as follows: $ _i =( S _i ; S _Oi )=(L _i , M _i , N _i ; P _i , Q _i , R _i ), S _i =(L _i , M _i , N _i )= ( B _i − C _j )/ $\Vert B_i-C_j\Vert$ is a unit vector specifying the direction of $ _i and S _Oi =(P _i , Q _i , R _i )= C _j ×( B _i − C _j )/ $\Vert B_i-C_j\Vert$ =( C _j × B _i )/ $\Vert B_i-C_j\Vert$ is a vector indicating the position of $ _i .

When the manipulator is singular, there is at least one undesired DOF, an instantaneous screw motion with a pitch h ^m . It can be obtained by using the following expression in [17]:

${\$}^{\text{m}}=\left|\begin{array}{cccccc}\in i& \in j& \in k& i& j& k\\ {\text{L}}_1& {\text{M}}_1& {\text{N}}_1& {\text{P}}_1& {\text{Q}}_1& {\text{R}}_1\\ {\text{L}}_2& {\text{M}}_2& {\text{N}}_2& {\text{P}}_2& {\text{Q}}_2& {\text{R}}_2\\ {\text{L}}_3& {\text{M}}_3& {\text{N}}_3& {\text{P}}_3& {\text{Q}}_3& {\text{R}}_3\\ {\text{L}}_4& {\text{M}}_4& {\text{N}}_4& {\text{P}}_4& {\text{Q}}_4& {\text{R}}_4\\ {\text{L}}_5& {\text{M}}_5& {\text{N}}_5& {\text{P}}_5& {\text{Q}}_5& {\text{R}}_5\end{array}\right|$ (2)

where $ ^m indicates the moving screw which is reciprocal to $ _i (i=1, 2, …, 5, 6). Similarly, $ ^m can also be expressed as a dual vector and expanded as follows, $ ^m =( $ ^m ; $m O)=L ^m i +M ^m j+N ^m k +∈(P ^m i +Q ^m j+R ^m k ). The pitch h ^m of the reciprocal screw $ ^m can be expressed by the following formula:

$h^m=(L^m{P}^m+M^m{Q}^m+N^m{R}^m)/(L^m{}^{\hspace{0.17em}2}+M^m{}^{\hspace{0.17em}2}+N^m{}^{\hspace{0.17em}2})$ (3)

3. Position-singularity analysis of the manipulator in a three-dimensional space

A moving reference frame P-X ^′ Y ^′ Z ^′ and a fixed one O-XYZ, respectively, are attached to the moving platform and the base of the manipulator, as shown in Fig. 1, where origin P is the geometric centre of the moving platform and O is the midpoint of the side, C₁C₂, of the base. The Cartesian coordinates of the moving platform are given by the position of point P with respect to the fixed frame, with components of (X, Y, Z), and the orientation of the moving platform is represented by three standard Z-Y-Z Euler angles (φ, θ, ψ). Furthermore, the geometric parameters of the parallel manipulator can be described as follows. The circumcircle radius of the base hexagon is R _a , and that of the moving platform is R _b . Meanwhile, β₀ denotes the central angle of the circumcircle of the base hexagon corresponding to the side C₅C₆, as shown in Fig. 1. The coordinates of the three vertices, B _i (i=1, 3, 5), of the moving platform are denoted by B _i ' with respect to the moving frame, and B _i in the fixed one. Similarly, C _i and A _j represent the coordinates of the vertices, C _i (i=1, 2, …, 5, 6) and A _j (j=1, 3, 5), of the base in the fixed frame.

Gosselin and Angeles [7] pointed out that the singularities of parallel manipulators could be classified according to three different types, i.e., the inverse kinematic singularity, the direct kinematic singularity and the architecture singularity. As to the second type of singularity, this is the most difficult to analyse and it has attracted a great deal of attention from researchers. As such, this paper will only discuss the direct kinematic singularity of this class of 3/6-Gough-Stewart parallel manipulators, which occurs when the determinant of the Jacobian matrix is equal to zero, i.e., det( J )=0. Expanding and factorizing the determinant of the Jacobian matrix, the position-singularity locus equation of the parallel manipulator can be written as follows:

$\begin{array}{l}{f}_1\hspace{0.17em}{Z}^3+f_2\hspace{0.17em}X\hspace{0.17em}{Z}^2+f_3\hspace{0.17em}Y\hspace{0.17em}{Z}^2+f_4\hspace{0.17em}{X}^{2}Z+f_5\hspace{0.17em}{Y}^{2}Z+f_6\hspace{0.17em}X\hspace{0.17em}Y\hspace{0.17em}Z+f_7\hspace{0.17em}{Z}^2+f_8\hspace{0.17em}{X}^2\\ +f_9\hspace{0.17em}{Y}^2+f_{10}\hspace{0.17em}X\hspace{0.17em}Y+f_{11}\hspace{0.17em}X\hspace{0.17em}Z+f_{12}\hspace{0.17em}Y\hspace{0.17em}Z+f_{13}\hspace{0.17em}Z+f_{14}\hspace{0.17em}X+f_{15}\hspace{0.17em}Y+f_{16}=0\end{array}$ (4)

Equation (4), a polynomial expression of degree three in the moving platform position parameters XYZ, represents the position-singularity locus of this class of 3/6-Gough-Stewart manipulators in a Cartesian space for a constant orientation (φ, θ, ψ). Coefficients of Equation (4), f _i (i=1, 2, …, 15, 16), are all functions of the geometric parameters, R _a , R _b and β₀, and orientation parameters (φ, θ, ψ).

Huang et al. [13,14] pointed out that when φ=±30°, ±90° or ±150°, ψ≠±30°, ±90° and ±150°, and θ≠0°, the position-singularity loci of this class of 3/6-Gough-Stewart manipulators all include a plane and a hyperbolic paraboloid; when φ=±30°, ±90°, or ±150°, ψ=±30°, ±90° or ±150°, and θ≠0°, the position-singularity loci of the manipulator includes three intersecting planes. One point to note is that further research shows that when φ≠±30°, ±90° and ±150°, ψ=±30°, ±90° or ±150°, and θ≠0°, the position-singularity loci of this class of 3/6-Gough-Stewart manipulators also include a plane and a hyperbolic paraboloid. This conclusion has not been mentioned elsewhere. For example, when ψ=90°, Equation (4) can be reduced as follows:

$\begin{array}{l}\text{(}2Z+R_b\sin \theta \text{)(}{a}_{11}{X}^2+a_{22}{Y}^2+a_{33}{Z}^2+2a_{23}YZ+\\ 2a_{31}ZX+2a_{12}XY+2a_{14}X+2a_{24}Y+2a_{34}Z+a_{44}\text{)}=0\end{array}$ (5)

Figure 2(a).

Position-singularity locus for the orientation (90°, 45°, 0°)

Figure 2(b).

Position-singularity locus for the orientation (90°, 45°, 30°)

Figure 2(c).

Position-singularity locus for the orientation (45°, −30°, 90°)

Figure 2(d).

Position-singularity locus for the orientation (60°,45°,45°)

Accordingly, it can be concluded that when θ≠0° and if either φ or ψ is equal to one of the six values ±30°, ±90° or ±150°, the position-singularity loci of this class of 3/6-Gough-Stewart manipulators all include a plane and a hyperbolic paraboloid; when θ≠0° and if both φ and ψ are equal to one of the above-mentioned six values, the position-singularity loci of the manipulator include three intersecting planes. Hence, the aforementioned six values of φ or ψ can be defined as special orientations of the manipulator, and general orientations of the manipulator can be described as the set of orientations, φ, θ, ψ, where neither φ nor ψ is equal to the aforementioned six values and θ≠0°. The properties of the singularity loci of the manipulator for general orientations will be discussed in the present paper.

In order to demonstrate the theoretical results just mentioned, graphical representations of the position-singularity locus of this class of the 3/6-Gough-Stewart manipulators for different orientations are given to illustrate the results, as shown in Fig. 2. The geometric parameters used here are given as R _a =2 m, R _b =3/2 m and β₀=90°. From Fig. 2, it can be clearly seen that the position-singularity loci of the parallel manipulator for different orientations are quite complex and various. Among them, the most complicated graphic of the position-singularity loci looks like a trifoliate surface, whose two branches are of the shape of a horn with one hole.

4. Position-singularity analysis of the manipulator based on a singularity-equivalent-mechanism

4.1. Position-Singularity Analysis of the Manipulator in θ-Plane

Based on the singularity kinematics principle proposed by Huang in 2002, Huang et al. [15] proposed a method called ‘singularity-equivalent-mechanism' by which the complicated singularity analysis of this class of 3/6-Gough-Stewart manipulators for special orientations is transformed into a simple direct position analysis of a planar singularity-equivalent-mechanism. It shows that cross sections of the cubic position-singularity locus of the 3/6-Gough-Stewart parallel manipulator in parallel θ-planes, on which the moving platform lies, are all quadratic expressions which include infinite hyperbolas. This conclusion is of great importance for the property identification of the position-singularity loci of the manipulator for special orientations. Similarly, in order to identify the characteristics of the position-singularity loci of this class of 3/6-Gough-Stewart Manipulators for general orientation, the position-singularity loci of the manipulator in parallel θ-planes will also be discussed using the method of singularity-equivalent-mechanism. As regards the θ-plane and position-singularity analysis of this class of 3/6-Gough-Stewart manipulators for special orientations, we refer the reader to the detailed discussion given in [13-15, 18].

For general orientations of the manipulator described in Section 3, since θ≠0° the moving platform is not parallel to the base. The θ-plane intersects the base plane at a line UVW, where points U, W and V are intersecting points between the θ-plane and three sides, A₃A₅, A₁A₅ and A₃A₁ of the base hexagon, respectively. In addition, for φ≠±30°, ±90° and ±150°, the intersecting line UVW is not parallel to any side of the triangle A₁A₃A₅, as shown in Fig. 3. Similarly, according to the singularity kinematics principle proposed by Huang in [13-15], a planar singularity-equivalent-mechanism can be constructed for the singularity analysis of this class of 3/6-Gough-Stewart manipulators for general orientations, as shown in Fig. 4. If we set another moving reference frame V-xy in the θ-plane, the coordinates of point P in the moving frame V-xy can be noted (x, y). The triangle B₁B₃B₅ is connected to the ground passing through three points W, V and U by three RPR kinematic chains, where R indicates the revolute pair and P the prismatic pair. The three slotted links L₁, L₂ and L₃, intersect at a common point C. In order to keep the three links intersecting at a common point, a concurrent kinematic chain, PRPRP, is used at point C. Hence, all configurations of the planar singularity-equivalent-mechanism satisfying the singularity kinematics principle are special configurations of this class of 3/6-Gough-Stewart manipulator, and the position-singularity locus equation of the manipulator for general orientations in a θ-plane can be obtained by analysing the direct kinematics of this planar singularity-equivalent-mechanism.

Figure 3.

The singular configuration of the manipulator for the general orientation

Figure 4.

The planar singularity-equivalent-mechanism for the general orientation

Owing to space limitations, we will not present the detailed process of the direct kinematics analysis of the planar singularity-equivalent-mechanism here, instead referring the reader to the detailed explanation given in [15, 18]. The position-singularity locus equation of this class of manipulators for general orientations in a θ-plane can be written as follows, after some rearrangements and factorizations, namely:

$bxy+c{y}^2+dx+ey+f=0$ (6)

Equation (6) is a quadratic polynomial expression with respect to xy and the maximum degree of the variable x is 1 and that of y is 2. The coefficients b, c, d, e, f, of Equation (6) are all functions of geometric parameters R _a , R _b and β₀, Euler angles (φ, ψ) and xV, and they are all independent of the Euler angle θ, where xV is a variable indicating the position on the θ-plane, i.e., the position of the moving platform of the manipulator.

4.2. Property Identification of the Position-Singularity Loci of the Manipulator on Parallel θ-Planes

The property of the position-singularity loci of this class of 3/6-Gough-Stewart manipulators on parallel θ-planes can be analysed by two invariants, D and δ, of Equation (6):

$D=\left|\begin{array}{ccc}\text{0}& b/2& d/2\\ b/2& c& e/2\\ d/2& e/2& f\end{array}\right|=-\frac{1}{4}(b^{\text{2}}f+d^{\text{2}}c-\mathit{bde})$ (7) $\delta =\left|\begin{array}{cc}0& b/2\\ b/2& c\end{array}\right|=-\frac{1}{4}{b}^2$ (8)

For any given set of geometric parameters R _a , R _b and β₀ and orientation parameters (φ, θ, ψ), further has research shown that D is a quartic expression while δ a quadratic expression with respect to the single variable xV. Generally, there are four real roots when D=0 and δ<0, and Equation (6) degenerates into a pair of intersecting lines. For the same reason, there is one real root of multiplicity two when δ=0 and D≠0, and Equation (6) degenerates into a parabola. With the exception of the aforementioned five special values of xV, D≠0 and δ<0 for general values of xV, therefore, Equation (6) indicates a set of hyperbolas. Hence, the following observation can be made, namely that when θ≠0°, the position-singularity loci of this class of 3/6-Gough-Stewart parallel manipulators for general orientations in parallel θ-planes are always quadratic expressions, which contain four pairs of intersecting lines, a parabola and infinite hyperbolas. Therefore, the θ-plane reflects characteristics of the position-singularity loci of the 3/6-Gough-Stewart manipulators; we call it a principal-section when analysing the position-singularity of the parallel manipulator.

4.3. Numerical Examples

As mentioned above, for any given set of geometric parameters R _a , R _b and β₀ and orientation parameters (φ, θ, ψ) when θ≠0°, the position-singularity loci of this class of 3/6-Gough-Stewart manipulators for general orientations in parallel principal-sections are always quadratic expressions including four pairs of intersecting lines, a parabola and infinite hyperbolas. In order to demonstrate the aforementioned theoretical results, a 3/6-Gough-Stewart manipulator will be studied numerically.

The intersecting line UVW passes through the point A₁ and then the two points W and V coincide with point A₁. In this case, xV ₁ = $\sqrt{3}$ R _a cos(β₀/2), Equation (6) degenerates into a pair of intersecting lines, as shown in Fig. 5:

$\begin{array}{l}[y-R_b\sin (\psi +{60}^{°})][(-\sqrt{3}\sin (\psi )+\cos (\psi ))x+\\ (\sqrt{3}\cos (\psi )+\sin (\psi ))y+R_b]=0\end{array}$ (9)

The first part of Equation (9) is:

$y-R_b\sin (\psi +{60}^{°})=0$ (10)

Figure 5.

The first pair of intersecting lines when x _V =x _V1

Figure 6(a).

B₅ does not coincide with A₁

Figure 6(b).

B₅ coincides with A₁

Equation (10) denotes a line parallel to the x-axis of the frame V-xy. Meanwhile, it can be proved that point B₅ is located on the base (Fig. 6(a)). The manipulator is always singular whatever the coordinate x is. The normal plane B₅A₃A₅ coincides with the base plane. Therefore, the intersecting line between two normal planes, B₅A₃A₅ and B₁A₁A₅, is A₁A₅, the intersecting line between normal planes, B₅A₃A₅ and B₃A₁A₃, is A₁A₃, the intersecting line between normal planes, B₁A₁A₅ and B₃A₁A₃, is A₁Q. So, A₁ is the intersecting point of three normal planes, B₅A₃A₅, B₁A₁A₅ and B₃A₁A₃. Considering that A₁ lies on the line UV which also lies on the mobile B₁B₃B₅, so point A₁ lies on the mobile B₁B₃B₅ as well. According to the singularity kinematics principle proposed in [13-15], the manipulator is singular.

In particular, when (x, y)=(R _b cos(ψ+60°), R _b sin(ψ+60°)), point B₅ coincides with point A₁ (Fig. 6(b)). In this case, the two planes determined by triangles B₁A₁A₅ and B₃A₁A₃ intersect with one common line A₁Q. Given that A₁Q passes through A₁ and B₅, it intersects with B₅A₅ and B₅A₃. Therefore, all the segments associated with the six extensible links of the manipulator intersect one common line A₁Q. It belongs to the first special-linear-complex singularity [13]. The undesired instantaneous motion of the manipulator is a pure rotation and A₁Q is the revolute axis of the instantaneous motion. It is also the singularity of Merlet 5b in [5].

This shows that the singularity of the point is of the first special-linear-complex singularity when B₅ coincides with A₁; and singularities of points lying in the line of Equation (10) are of the general-linear-complex singularity [13] when B₅ does not coincide with A₁. In these cases, the corresponding twists and its pitches can be calculated by Equations (2) and (3):

${\$}_1^m=\text{(}0.876,\hspace{0.17em}-0.109,\hspace{0.17em}0.469\text{;\hspace{0.17em}}0,\hspace{0.17em}-1.150,\hspace{0.17em}-0.267\text{)}$ (11) $h_1^m=-3.27998287\times {10}^{-20}=0$ (12) ${\$}_2^m=\text{(}0.876,\hspace{0.17em}-0.109,\hspace{0.17em}0.469\text{;\hspace{0.17em}}0,176,\hspace{0.17em}-0.845,\hspace{0.17em}-0.877\text{)}$ (13) $h_2^m=-0.1653$ (14)

The second part of Equation (9) denotes another straight line. Singularities corresponding to points lying in this line are all of the general-linear-complex singularity. It is also the singularity of the type Merlet 5a in [5], with the following corresponding twist and pitch:

${\$}_3^m=\text{(}-0.794,\hspace{0.17em}-0.433,\hspace{0.17em}-0.426\text{;\hspace{0.17em}}0,\hspace{0.17em}1.371,\hspace{0.17em}-0.709\text{)}$ (15) $h_3^m=-0.7093$ (16)

Figure 7.

The second pair of intersecting lines when x _V =x _V2

Similarly, UV passes through point A₃. In this case xV ₂ =0, two points U and V coincide with point A₃. The Equation (6) degenerates into a pair of intersecting lines, as shown in Fig. 7:

$\begin{array}{l}[(y+R_b\sin (\psi )]\\ [x\cos (\psi )+y\sin (\psi )-R_b/2]=0\end{array}$ (17)

The first part of Equation (17) indicates a straight line parallel to the x-axis. In particular, when(x, y)=(−R _b cos(ψ), -R _b sin(ψ)), B₁ coincides with point A₃, the singularity of this point is the first special-linear-complex singularity and the instantaneous motion is a pure rotation with h ^m =0. When B₁ does not coincide with A₃, the singularities of points lying in this straight line are the general-linear-complex singularity and its instantaneous motion is a twist with h ^m ≠0.

The second part of Equation (17) denotes another straight line. The singularities of points lying in this straight line are all the general-linear-complex singularity with h ^m ≠0.

Figure 8.

The third pair of intersecting lines when x _V =x _V3

Figure 9.

The fourth pair of intersecting lines when x _V =x _V4

Figure 10.

A parabola when x _V =x _V5

Figure 11.

A hyperbola when x _V =1

Similarly, when xV is equal to one of the following three values,

$\begin{array}{l}{x}_{v3}=\sqrt{3}{R}_a\cos \text{(}{\beta }_0/2\text{)sin(}\varphi +{\text{30}}^{\circ }\text{)/sin(}\varphi -{\text{30}}^{\circ }\text{),}\\ x_{v4}=(-1+2\cos (2\varphi ))(R_b\cos \varphi -2R_a\cos ({\beta }_0/2)\cos \psi )/\\ (2(\sqrt{3}\sin \varphi -\cos \varphi )\sin (\varphi +\psi )),\\ x_{v5}=R_a\cos ({\beta }_0/2)\cos \psi (\cos \varphi +\sqrt{3}\sin \varphi )/\sin (\varphi +\psi )\end{array}$

Equation (6) degenerates into two pairs of intersecting lines and a parabola, as shown in Fig. 8 and Fig. 10, respectively. With the exception of the aforementioned five special values of xV, D≠0 and δ<0, therefore, Equation (6) indicates a set of hyperbolas, as shown in Fig. 11 and Fig. 12, respectively.

Figure 12.

A hyperbola when x _V =-1

4.4. Property Identification of the Position-Singularity Loci of the Manipulator in Three-Dimensional Space

Based on the analyses described above, it can be safely concluded that the position-singularity loci of this class of 3/6-Gough-Stewart manipulators in parallel principal-sections contain infinite hyperbolas, four pairs of intersecting lines and a parabola when θ≠0°. From analytic geometry, there are five different kinds of quadric surfaces with hyperbola sections. They are a hyperbolic cylinder, a hyperbolic paraboloid, a hyperboloid of one sheet, a hyperboloid of two sheets and a conic surface. However, none of them can contain hyperbolas, a parabola and four pairs of intersecting lines simultaneously. Therefore, the position-singularity locus equation of this class of 3/6-Gough-Stewart manipulators for general orientations in a three-dimensional space is a special irresolvable polynomial expression of degree three whose cross sections in parallel principal-sections include a parabola, four pairs of intersecting lines and infinite hyperbolas.

5. Conclusion

According to the analyses described above, the property of the position-singularity loci of this class of the 3/6-Gough-Stewart manipulators for all orientations, including special orientations and general orientations, can be finally concluded as follows:

1. When θ=0°, (φ+ψ)=±90°, for the 3/6-Gough-Stewart manipulator, it is the singularity proposed by Fichter [3]. It belongs to the general-linear-complex singularity.

2. When θ=0°, Z=0, the moving platform and the base one are coincident. In this special configuration, the manipulator has three undesired DOFs: two rotational freedoms and one translational freedom. They belong to the first and the second special-linear-complex singularity, respectively.

3. When θ≠0°, φ=±30°, ±90° or ±150°, and ψ≠±30°, ±90° and ±150°, the position-singularity loci of the manipulator include a plane and a hyperbolic paraboloid.

4. When θ≠0°, φ≠±30°, ±90° and ±150°, and ψ=±30°, ±90° or ±150°, the position-singularity loci of the manipulator also include a plane and a hyperbolic paraboloid.

5. When θ≠0°, φ=±30°, ±90° or ±150°, and ψ=±30°, ±90° or ±150°, position-singularity loci of the manipulator include three intersecting planes.

6. When θ≠0° and neither φ nor ψ is equal to the following six values ±30°, ±90° or ±150°, the position-singularity locus equation of the manipulators in a three-dimensional space is a special irresolvable polynomial expression of degree three whose cross sections in parallel principal-sections include a parabola, four pairs of intersecting lines and infinite hyperbolas.