Unified Singularity Modeling and Reconfiguration of 3rT PS Metamorphic Parallel Mechanisms with Parallel Constraint Screws

Abstract

This paper presents a unified singularity modeling and reconfiguration analysis of variable topologies of a class of metamorphic parallel mechanisms with parallel constraint screws. The new parallel mechanisms consist of three reconfigurable rTPS limbs that have two working phases stemming from the reconfigurable Hooke (rT) joint. While one phase has full mobility, the other supplies a constraint force to the platform. Based on these, the platform constraint screw systems show that the new metamorphic parallel mechanisms have four topologies by altering the limb phases with mobility change among 1R2T (one rotation with two translations), 2R2T, and 3R2T and mobility 6. Geometric conditions of the mechanism design are investigated with some special topologies illustrated considering the limb arrangement. Following this and the actuation scheme analysis, a unified Jacobian matrix is formed using screw theory to include the change between geometric constraints and actuation constraints in the topology reconfiguration. Various singular configurations are identified by analyzing screw dependency in the Jacobian matrix. The work in this paper provides basis for singularity-free workspace analysis and optimal design of the class of metamorphic parallel mechanisms with parallel constraint screws which shows simple geometric constraints with potential simple kinematics and dynamics properties.

1. Introduction

Parallel mechanisms [1] have potential advantages on load-carrying capacity, good positioning accuracy, and low inertia, resulting in great interests in mechanism research aiming to produce high performance machines in applications in industry and robotics world. In order to keep these advantages of traditional parallel mechanisms [2, 3] but add additional adaptability, metamorphic parallel mechanisms (MPMs) [4] were developed. MPMs are a class of mechanisms that possess adaptability and reconfigurability to change permanent finite mobility based on topological structure change. In this paper, a class of metamorphic parallel mechanisms with parallel constraint screws are introduced and a unified singularity modelling is provided. This will be useful in optimally designing this class of parallel mechanisms which shows simple geometric constraints with potential simple kinematics and dynamics properties for practical applications.

The concept of metamorphic parallel mechanism came from metamorphic mechanisms originated from the study of decorative carton folds and reconfigurable packaging [5]. Much research work has been devoted to this area due to the novel property of reconfiguration and mobility change of metamorphic mechanisms. These include new methods to synthesize mechanisms with reconfiguration, like variable position parameters for kinematotropic linkages [6] and kinematotropic parallel mechanisms [7], metamorphic ways of changing the topological structures of a mechanism [8], screw theory based synthesis of parallel mechanisms with multiple operation modes [9], variable topology joints for variable topology mechanisms [10], and methodology for synthesis based on biological modeling and genetic evolution [11]. Following this, some research interests move to the topology description of various mobility-configurations of metamorphic mechanisms, including matrix operations [12], joint code [13], joint-gene based topological representation [14], lie displacement subgroup [15], and operation space with motion compatibility representation [16]. Other works are related to applications of metamorphic mechanisms, like metamorphic underwater vehicle [17] and metamorphic multifingered hand with an articulated palm [18]. Recently, some work has been done on metamorphic parallel mechanisms. Based on a reconfigurable Hooke (rT) joint, two types of MPMs were introduced [4] and detailed topology reconfiguration of an MPM with mobility change from 1 to 6 was investigated [19]. A general construction method for MPMs was introduced using screw theory [20] and a metamorphic parallel mechanism with ability of performing phase change and orientation switch was also proposed in [21]. In [22], motion planning for a parallel mechanism with reconfigurable configurations was studied based on lockable revolute joints.

With previous focus on the synthesis and topology representation, little work has been done on the variation of kinematics [23], dynamics, singularity, and workspace of MPMs in their reconfiguration. In this paper, a class of metamorphic parallel mechanism based on a reconfigurable rTPS limb [24] will be introduced with another focus on the unified singularity modeling and singularity change to cover all their reconfigurable topologies. In parallel mechanism research, singularity analysis [25, 26] is an important topic as singularities will make the mechanism dysfunctional and uncontrollable which should be avoided and considered before parameter design for real applications. Many methods have been proposed for singularity analysis which mainly includes Jacobian-determinant-based numerical methods [27] and Jacobian-rank-based analytical models [25]. Since this paper does not focus on a parallel mechanism with fixed dimensions, the latter one will be suitable. To investigate the Jacobian rank degeneracy, screw theory [28, 29], line geometry, and Grassmann-Cayley algebra [30, 31] were used to model the constraint and actuation forces to investigate their dependency. Based on Grassmann geometry and linear varieties from [32], Merlet [33] successfully found all the singularities of the 6-3 Gough-Stewart parallel mechanism. This method was then expanded by Hao and McCarthy [30] using screw theory with systematic analysis of possible dependent screws. Considering the variable topologies of the MPMs in this paper, screw theory shows a good way to formulize a unified Jacobian matrix [34] to include both geometric constraints and actuation constraints with the change between them in the reconfiguration while the size is kept 6 by 6.

The paper is arranged as follows. Section 2 introduces the two working phases of the reconfigurable rTPS limb with their constraint screws. The new metamorphic parallel mechanisms with parallel constraint screws are presented in Section 3 with mobility analysis, geometric design conditions, and some special topologies. Following this, Section 4 analyzes the topology reconfiguration and mobility change. Section 5 shows possible actuation scheme for variable topologies. Based on these, Section 6 proposes the unified singularity modeling and various singularity configurations with different mechanism topologies. Conclusions are made in Section 7.

2. Two Phases of the Reconfigurable rTPS Limb

The reconfigurable rTPS limb consists of a reconfigurable Hooke (rT) joint, a prismatic joint, and a spherical joint. The reconfigurability of this limb stems from the configuration change of the rT joint which has two rotational degrees of freedom (DOFs) about two perpendicularly intersecting rotational axes (radial axis and bracket axis) as in Figure 1. A grooved ring is used to house the radial axis and make it have the ability of altering its direction by rotating and fixing freely along the groove. This allows the radial rotation axis change with respect to the limb, resulting in two typical phases of the rTPS limb as in Figure 1. While in Figure 1(a), the radial axis is perpendicular to the limb (prismatic joint) which is denoted as (rT)₁PS, it is collinear with the limb (prismatic joint) passing through the spherical joint center in Figure 1(b) and the limb phase is symbolized as (rT)₂PS.

Figure 1:

Two phases of the rTPS limb.

Set a limb coordinate system ¹o¹x¹y¹z at the rT joint center with ¹x axis collinear with the bracket axis and ¹y axis perpendicular to the bracket surface as in Figure 1(a); the twist system of the (rT)₁PS limb is given as

$\left\{{}^1{\mathit{S}}_1\right\}=\left\{\begin{array}{c}{}^1{\mathit{S}}_{11}=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\end{array}\right]\\ {}^1{\mathit{S}}_{12}=\left[\begin{array}{cccccc}0& -s\alpha & c\alpha & 0& 0& 0\end{array}\right]\\ {}^1{\mathit{S}}_{13}=\left[\begin{array}{cccccc}0& 0& 0& -s\beta & c\beta c\alpha & c\beta s\alpha \end{array}\right]\\ {}^1{\mathit{S}}_{14}=\left[\begin{array}{cccccc}0& c\alpha & s\alpha & 0& ls\beta s\alpha & -ls\beta c\alpha \end{array}\right]\\ {}^1{\mathit{S}}_{15}=\left[\begin{array}{cccccc}1& 0& 0& 0& lc\beta s\alpha & -lc\beta c\alpha \end{array}\right]\\ {}^1{\mathit{S}}_{16}=\left[\begin{array}{cccccc}0& -s\alpha & c\alpha & lc\beta & ls\beta c\alpha & ls\beta s\alpha \end{array}\right]\end{array}\right\},$

(1)

where the first two twists are generated from the rT joint, the third is generated from the prismatic joint, and the last three are generated from the spherical joint. β is the angle between the limb ( ${}^1{\mathit{S}}_{13}$ ) and its projection on plane ¹y¹o¹z passing through rT joint center and perpendicular to the bracket axis (¹x); α is the angle between axis ¹y and the limb projection on plane ¹y¹o¹z. l is the distance between the rT joint centre and the spherical joint centre. Labels cα and sα denote Cosine(α) and Sin(α), respectively. In the twist notation ${}^1{\mathit{S}}_{ij}^{}$ , the first subscript i denotes the limb number, the second subscript j denotes the joint number within the limb, and the leading superscript indicates the local frame.

The six screws in (1) form a six-system [35] and there is no reciprocal screw, showing that the (rT)₁PS limb has six degrees of freedom (DOFs) and does not supply constraint to the platform connected to it.

For the (rT)₂PS limb as in Figure 1(b), radial axis of the rT joint is collinear with the prismatic joint passing through the spherical joint center. Thus, the spherical joint center A cannot move out of the plane ¹y¹o¹z. Based on the coordinate system in Figure 1(b), the twist system of the (rT)₂PS limb can be given as

$\left\{{}^1{\mathit{S}}_1\right\}=\left\{\begin{array}{c}{}^1{\mathit{S}}_{11}=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\end{array}\right]\\ {}^1{\mathit{S}}_{12}=\left[\begin{array}{cccccc}0& -c\alpha & -s\alpha & 0& 0& 0\end{array}\right]\\ {}^1{\mathit{S}}_{13}=\left[\begin{array}{cccccc}0& 0& 0& 0& c\alpha & s\alpha \end{array}\right]\\ {}^1{\mathit{S}}_{14}=\left[\begin{array}{cccccc}0& c\alpha & s\alpha & 0& 0& 0\end{array}\right]\\ {}^1{\mathit{S}}_{15}=\left[\begin{array}{cccccc}1& 0& 0& 0& ls\alpha & -lc\alpha \end{array}\right]\\ {}^1{\mathit{S}}_{16}=\left[\begin{array}{cccccc}0& -s\alpha & c\alpha & l& 0& 0\end{array}\right]\end{array}\right\}.$

(2)

It is clear that twists ${}^1{\mathit{S}}_{12}$ and ${}^1{\mathit{S}}_{14}$ are the same and the six twists form a five-system. Thus, there is one reciprocal screw to (2) in the limb constraint system as follows:

$\left\{{}^1{\mathit{S}}_1^r\right\}={}^1{\mathit{S}}_1^r=\left[\begin{array}{cccccc}1& 0& 0& 0& ls\alpha & -lc\alpha \end{array}\right].$

(3)

This gives a constraint force acting along a line passing through the spherical joint centre with a direction parallel to the bracket axis of the rT joint. Thus, the (rT)₂PS limb has five DOFs, one less than the (rT)₁PS phase.

When constructing parallel mechanisms with the rTPS limbs, the mechanisms will have ability of mobility change by altering the rTPS limbs into these two phases. In the following, the (rT)₂PS limb phases will be used by connecting the rT joints to the base with parallel bracket axis and the spherical joints to the moving platform. Then altering the limbs into (rT)₁PS phases will generate new mechanism topologies, which gives a class of metamorphic parallel mechanisms with parallel constraint screws.

3. 3rTPS MPMs with Parallel Constraint Screws

3.1. Mobility Analysis

From (3), it can be seen that (rT)₂PS limb gives one constraint force to the platform with the direction parallel to its bracket axis. Thus, by arranging the direction of the bracket axis of the rT joint, the constraint to the platform can be represented. In this paper, the rT joints are connected to the base with parallel bracket axes, leading to parallel constraint forces expressed by screws on the platform as in Figure 2.

Figure 2:

The 3(rT)₂PS with parallel constraint screws.

Let points A _i and B _i denote the spherical joint center and the rT joint center in limb i (i = 1, 2, 3), respectively. Locate a global coordinate system oxyz at the center (B₁) of the rT joint in limb 1 with x axis collinear with the bracket axis and y axis perpendicular to the bracket plane. Let a _i and b _i denote the vectors of points A _i and B _i in the coordinate system oxyz, and let l _i be the distance between the spherical joint center A _i and the rT joint center B _i .

The constraint system of the 3(rT)₂PS parallel mechanism in Figure 2 can be given as

$\left\{{\mathit{S}}^r\right\}=\left\{\begin{array}{c}{\mathit{S}}_{\mathrm{1}}^r=\left[\begin{array}{cc}\mathit{n}& {\mathit{a}}_{\mathrm{1}}\times \mathit{n}\end{array}\right]\\ {\mathit{S}}_{\mathrm{2}}^r=\left[\begin{array}{cc}\mathit{n}& {\mathit{a}}_{\mathrm{2}}\times \mathit{n}\end{array}\right]\\ {\mathit{S}}_{\mathrm{3}}^r=\left[\begin{array}{cc}\mathit{n}& {\mathit{a}}_{\mathrm{3}}\times \mathit{n}\end{array}\right]\end{array}\right\}=\left\{\begin{array}{c}\left[\begin{array}{cccccc}\mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& l_{\mathrm{1}}s{\alpha }_{\mathrm{1}}& -l_{\mathrm{1}}c{\alpha }_{\mathrm{1}}\end{array}\right]\\ \left[\begin{array}{cccccc}\mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& b_{\mathrm{2}z}+l_{\mathrm{2}}s{\alpha }_{\mathrm{2}}& -b_{\mathrm{2}y}-l_{\mathrm{2}}s{\alpha }_{\mathrm{2}}\end{array}\right]\\ \left[\begin{array}{cccccc}\mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& b_{\mathrm{3}z}+l_{\mathrm{3}}s{\alpha }_{\mathrm{3}}& -b_{\mathrm{3}y}-l_{\mathrm{3}}s{\alpha }_{\mathrm{3}}\end{array}\right]\end{array}\right\},$

(4)

where n = (1, 0, 0) is the direction of the bracket axes, a _i = b _i + l _i (0, cα _i , sα _i ) = (b _ix , b _iy , b _iz ) + l _i (− sβ _i , cβ _i cα _i , cβ _i sα _i ), α _i is the angle between limb i and the line passing through rT joint center B _i with direction parallel to y axis, and β _i is the angle between the limb and its projection on yoz plane passing through rT joint center and perpendicular to the bracket axis; here β _i = 0 in the (rT)₂PS as in (4).

By taking reciprocal screws to (4), motion screws of the mechanism can be obtained as

$\left\{{\mathbf{S}}^r\right\}=\left\{\begin{array}{c}\left[\begin{array}{cccccc}\mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right]\\ \left[\begin{array}{cccccc}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}& \mathrm{0}\end{array}\right]\\ \left[\begin{array}{cccccc}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right]\end{array}\right\},$

(5)

which represent three DOFs with one rotation about x axis and two translations along y and z axes.

Thus, the 3(rT)₂PS parallel mechanism with parallel constraint screws has mobility three with a rotation parallel to the screws and two translations perpendicular to the screws.

3.2. Geometric Conditions of the Mechanism Assembly

Based on the geometry of (rT)₂PS limb in Figure 1(b), it can be obtained that the spherical joint center cannot move out of the plane ∑ passing through the limb and perpendicular to the bracket axis ( n ) of the rT joint. This can be used to determine the assembly condition of the 3(rT)₂PS parallel mechanism in Figure 2. For each (rT)₂PS limb, there is a constraint plane ∑ for the spherical joint center A and the three constraint planes for the three limbs are parallel. However, these three constraint planes representing the limb locations cannot be arranged arbitrarily. The intrinsic constraint for this can be investigated by fixing constraint planes for limb 1 and limb 2 first and then identifying the conditions for limb 3. When giving spherical joint centers A₁ and A₂, the geometric constraints for the third spherical joint center A₃ can be obtained by intersecting constraint plane ∑3 with a circle centered at point A₃₀ with radius A₃A₃₀ as in Figure 3. The circle is intersecting of two spheres centered at points A₁ and A₂ with radii A₁A₃ and A₂A₃, respectively. A₃₀ is the projection point of A₃ on line A₁A₂.

Figure 3:

Location condition of constraint plane ∑3.

In Figure 3, β₁₃ is the angle between A₁A₃ and norm n ₁. It is dependent on the location of plane ∑3 as in Figure 3. When plane ∑3 is tangible with the constraint circle as in location ∑3′ or ∑3^′′, angle β₁₃ has maximum and minimum values β_{13max ⁡} and β_{13min ⁡}, respectively. Then there is

$\begin{array}{c}{l}_{\mathrm{13}}\mathbf{c}\mathbf{o}\mathbf{s}{\beta }_{\mathrm{13}\mathbf{m}\mathbf{a}\mathbf{x}}\le d_{\mathrm{3}}-d_{\mathrm{1}}=l_{\mathrm{13}}\mathbf{c}\mathbf{o}\mathbf{s}{\beta }_{\mathrm{13}}\le l_{\mathrm{13}}\mathbf{c}\mathbf{o}\mathbf{s}{\beta }_{\mathrm{13}\mathbf{m}\mathbf{i}\mathbf{n}},\\ {\beta }_{\mathrm{13}\mathbf{m}\mathbf{a}\mathbf{x}}={\phi }_{\mathrm{1}}+{\beta }_{\mathrm{12}},\\ {\beta }_{\mathrm{13}\mathbf{m}\mathbf{i}\mathbf{n}}={\phi }_{\mathrm{1}}-{\beta }_{\mathrm{12}},\end{array}$

(6)

where ϕ₁ is the platform angle A₂A₁A₃ and β₁₂ is the angle between A₁A₂ and norm n ₁.

Thus, constraint plane ∑3 should be located at ∑3′ or ∑3^′′ or between them as in Figure 3. When it is at ∑3′ or ∑3^′′, there is one intersecting point ( $A_{3^{\prime }}$ or $A_{3^{\prime \prime }}$ ) and there are two intersecting points (A₃ and A′₃) when plane ∑3 is between the two extreme locations. From Figure 3, it can also be seen that location (d₃) of plane ∑3 determines the orientation of the platform about line A₁A₂ when points A₁ and A₂ are fixed.

Here, the analysis is based on fixing points A₁ and A₂ by giving three actuation inputs to limb 1 and limb 2. But in mechanism actuation, normally the three inputs are given to the three limbs with one each to determine the platform displacement, which will be discussed in Section 5.

3.3. Special Topologies

When considering particular arrangements of the base and platform joints, the 3(rT)₂PS parallel mechanism will have different topologies and some special cases are identified below.

A special one is that the three spherical joint centers (A₁, A₂, and A₃) on the platform are in line as in Figure 4(a). The constraint screws in (4) become

$\left\{{\mathit{S}}^r\right\}=\left\{\begin{array}{c}{\mathit{S}}_{\mathrm{1}}^r=\left[\begin{array}{cc}\mathit{n}& {\mathit{a}}_{\mathrm{1}}\times \mathit{n}\end{array}\right]\\ {\mathit{S}}_{\mathrm{2}}^r=\left[\begin{array}{cc}\mathit{n}& ({\mathit{a}}_{\mathrm{1}}+l_{\mathrm{12}}{\mathit{m}}_A)\times \mathit{n}\end{array}\right]\\ {\mathit{S}}_{\mathrm{3}}^r=\left[\begin{array}{cc}\mathit{n}& ({\mathit{a}}_{\mathrm{1}}+l_{\mathrm{13}}{\mathit{m}}_A)\times \mathit{n}\end{array}\right]\end{array}\right\}=\left\{\begin{array}{c}\left[\begin{array}{cc}\mathit{n}& {\mathit{a}}_{\mathrm{1}}\times \mathit{n}\end{array}\right]\\ \left[\begin{array}{cc}\mathbf{0}& l_{\mathrm{12}}{\mathit{m}}_A\times \mathit{n}\end{array}\right]\\ \left[\begin{array}{cc}\mathbf{0}& l_{\mathrm{13}}{\mathit{m}}_A\times \mathit{n}\end{array}\right]\end{array}\right\},$

(7)

where l_1i is the distance between points A₁ and A _i (i = 2, 3) which are all on the line with unit vector m _A . Thus the last two constraint moments in (14) constrain the same rotation between line m _A and norm n . One of limb 2 and limb 3 becomes redundant, resulting in a local rotational DOF about line m _A , and the mechanism has two translations and two rotations.

Figure 4:

Two special topologies.

Another special topology is that the three rT joint centers (B₁, B₂, and B₃) on the base are in line with unit vector m _B as in Figure 4(b); the platform still has two translations and one rotation as in (5). However, when m _B = n = (1, 0, 0), the three prismatic joints cannot be chosen as actuations at the same time, which will be discussed in Section 5.

Two other special topologies of the 3(rT)₂PS with parallel constraint screws can be obtained when locating the rT joint center B₂ on yoz plane with b_2x = 0 as in Figure 5(a) and setting both rT joint centers (B₂ and B₃) on yoz plane with b_2x = b_3x = 0 as in Figure 5(b). Both topologies have two translations and one rotation and the topology in Figure 5(b) is a planar parallel mechanism.

Figure 5:

Three more special topologies.

One more special topology can be obtained by setting line B _i B _j parallel to n with distance l _ij between two spherical joint centers A _i A _j equal to the distance d _ij between two rT joint centers B _i B _j as in Figure 5(c) in which l₂₃ = d₂₃. In this case, the two constraint screws in limb 2 and limb 3 are collinear and dependent. The platform will have two independent constraint screws with four reciprocal motion screws including two rotations and two translations. However, the geometric constraint l₂₃ = d₂₃ makes line A₂A₃ constantly parallel with x axis and the platform can only rotate about line A₂A₃. Thus, the mechanism still has one rotation and two translation DOFs, which is a structure singularity case.

4. Topology Reconfiguration

Altering the (rT)₂PS limbs in the previous 3(rT)₂PS parallel mechanisms into the phase (rT)₁PS will result in various new mechanism topologies with increased mobility. After changing the phase of one limb, all the 3(rT)₂PS parallel mechanisms become the topology 2(rT)₂PS-1(rT)₁PS in Figure 6(a) that has two parallel constraint screws following the first two in (4). Thus the 2(rT)₂PS-1(rT)₁PS has one constraint less and will have one DOF more than the 3(rT)₂PS. Based on the constraint screw analysis, the 2(rT)₂PS-1(rT)₁PS parallel mechanism has four DOFs with two translations and two rotations (2R2T).

Figure 6:

The 2(rT)₂PS-1(rT)₁PS topologies.

A special case is the 3(rT)₂PS-e in Figure 5(c). When changing limb 2 or limb 3 from phase (rT)₂PS to (rT)₁PS in Figure 5(c), the mechanism changes to the case in Figure 6(a). However, when changing the phase of limb 1, the mechanism becomes topology 2(rT)₂PS-1(rT)₁PS in Figure 6(b) which has mobility three with one rotation and two translations (1R2T). Thus, the mechanism does not change mobility due to the structure singularity.

When further changing one more limb phase, both the topologies in Figure 6 change to the same topology 1(rT)₂PS-2(rT)₁PS as in Figure 7(a), which has one constraint screw that limited the translation along n . Thus, this mechanism has five DOFs with three rotations and two translations (3R2T). It can be noticed that the 2(rT)₂PS-1(rT)₁PS in Figure 7(b) changes its mobility from 3 to 5 in this reconfiguration.

Figure 7:

Topologies 1(rT)₂PS-2(rT)₁PS and 3(rT)₁PS.

When changing the third limb to phase (rT)₁PS, the mechanism becomes another topology 3(rT)₁PS as in Figure 7(b) that does not have any constraint screw and has full mobility 6.

One topology that should be mentioned is the 3(rT)₂PS-d in Figure 5(b), in which all three limbs are constrained in one plane and it is a planar parallel mechanism. By changing the limb phases one by one, the topologies become those in Figure 6(a) and the two in Figure 7. Thus, a planar parallel mechanism becomes a spatial parallel mechanism while the mobility changes from 3 to 6.

5. Actuation Scheme for the Reconfigurable Topologies

Actuation inputs of a parallel mechanism work with limb constraints to determine the platform position and orientation. In the (rT)₁PS limb, the rotation about the bracket axis and the radial axis and the translation along the prismatic joint can be selected as actuated joints. Based on Figure 8 and the limb motion screws in (1), these three inputs will result in constraint screws as

$\begin{array}{c}{}^1{\mathit{S}}_{rTba}^r=\left[\begin{array}{cccccc}0& -s\alpha & c\alpha & lc\beta & lc\alpha s\beta & ls\alpha s\beta \end{array}\right],\\ {}^1{\mathit{S}}_{rTra}^r=\left[\begin{array}{cccccc}c\beta & c\alpha s\beta & s\alpha s\beta & 0& ls\alpha & -lc\alpha \end{array}\right],\\ {}^1{\mathit{S}}_{Pa}^r=\left[\begin{array}{cccccc}-s\beta & c\alpha c\beta & s\alpha c\beta & 0& 0& 0\end{array}\right],\end{array}$

(8)

where ¹S _rTba ^r , a constraint force passing through the spherical joint center and perpendicular to both the bracket axis and the limb, is the actuation force from the bracket axis rotation. ¹S _rTra ^r is the input constraint force from the radial axis rotation which passes through the spherical joint center and perpendicular to both the radial axis and the limb. ¹S _Pa ^r is the actuation force from the prismatic joint and is along the limb.

Figure 8:

Actuation forces of the rTPS limb.

When altering the (rT)₁PS limb to phase (rT)₂PS, the radial axis is along the limb and the radial axis rotation cannot be an actuation input but a geometric force is produced as in (3). Based on the analysis in Section 2, the (rT)₂PS phase can be taken as a special case of the (rT)₁PS phase with the radial axis rotation angle β = 0. This can also be explained by the variation of the actuation forces. When giving β = 0 in (8), the actuation force from the radial axis becomes exactly the same with the geometric constraint in (3) generated in the (rT)₂PS phase. Thus, the rotation about the bracket axis and the translation of the prismatic joint can be chosen as the actuation inputs in the (rT)₂PS limb with expression in (8) using β = 0, which is one choice less than the (rT)₁PS limb but with equivalent constraints considering the geometric constraint.

The limb actuation analysis gives the base for the mechanism actuation scheme. The 3(rT)₂PS with parallel constraint screws has three DOFs and needs three actuation inputs. Based on (8) with β = 0, each limb gives two possible actuation choices and there are totally six for the three limbs. Taking three of the six actuations should constrain the three DOFs of the mechanism. For the 3(rT)₂PS parallel mechanism in Figure 2, when choosing two prismatic joint inputs in limb 1 and limb 2 with a bracket rotation input in limb 3, the input constraints for the platform are

$\left\{{\mathit{S}}_a^r\right\}=\left\{\begin{array}{c}{\mathit{S}}_{\mathrm{1}Pa}^r=\left[\begin{array}{cc}{\mathit{u}}_{\mathrm{1}}& \mathrm{0}\end{array}\right]\\ {\mathit{S}}_{\mathrm{2}Pa}^r=\left[\begin{array}{cc}{\mathit{u}}_{\mathrm{2}}& {\mathit{b}}_{\mathrm{2}}\times {\mathit{u}}_{\mathrm{2}}\end{array}\right]\\ {\mathit{S}}_{\mathrm{3}rTba}^r=\left[\begin{array}{cc}\mathit{n}\times {\mathit{u}}_{\mathrm{3}}& {\mathit{a}}_{\mathrm{3}}\times (\mathit{n}\times {\mathit{u}}_{\mathrm{3}})\end{array}\right]\end{array}\right\},$

(9)

where u _i is the unit vector along limb i.

The three input forces in (9) with the three geometric constraint forces in (4) form a six-system, showing that the selected actuation inputs can fix the required position and orientation of the platform. Following this way, actuations for the 3(rT)₂PS parallel mechanisms including the special topologies can be obtained and concluded as follows.

1. Any three of the six inputs (the prismatic translation and bracket rotation in each limb) can be actuations for the general 3(rT)₂PS in Figure 2; the 3(rT)₂PS-a, 3(rT)₂PS-b, 3(rT)₂PS-c, and 3(rT)₂PS-d except the case that the three rT joint centers are on x axis of the 3(rT)₂PS-b that cannot have three prismatic joints as the actuation.

2. The 3(rT)₂PS-a needs three of the six inputs with an extra for the local rotation of the platform.

3. At least one of the three inputs should be from limb 2 for actuations of the 3(rT)₂PS-e parallel mechanism.

When altering the (rT)₂PS limb to phase (rT)₁PS, one more actuation choice appears as the rotation about the radial axis as in (8). Based on this and the above analysis, the actuation scheme rules are as follows.

1. At least one of the actuations should be from the (rT)₁PS limb in the reconfigured topologies 2(rT)₂PS-1(rT)₁PS in Figure 6 and 1(rT)₂PS-2(rT)₁PS and 3(rT)₁PS in Figure 7.

2. At least two should be from the (rT)₁PS limb in the reconfigured topology 2(rT)₂PS-1(rT)₁PS in Figure 6(b).

6. Unified Singularity Modeling and Analysis

The infinitesimal twist of the moving platform of the 3(rT)PS parallel mechanism can be written as the linear combination of instantaneous twists of each limb:

${\mathbf{S}}_G={\dot{\alpha }}_i{\mathbf{S}}_{i\mathrm{1}}+{\dot{\beta }}_i{\mathbf{S}}_{i\mathrm{2}}+{\dot{l}}_i{\mathbf{S}}_{i\mathrm{3}}+{\dot{\varphi }}_{i\mathrm{4}}{\mathbf{S}}_{i\mathrm{4}}+{\dot{\varphi }}_{i\mathrm{5}}{\mathbf{S}}_{i\mathrm{5}}+{\dot{\varphi }}_{i\mathrm{6}}{\mathbf{S}}_{i\mathrm{6}}\hspace{1em}\left(i=\mathrm{1,2},\mathrm{3}\right),$

(10)

where S _G represents the infinitesimal twist of the moving platform, S _ij (j = 1, 2, 3, 4, 5, 6) denotes the unit screw of the jth 1-DOF joint in limb i, ${\dot{l}}_i$ is the distance rate of the prismatic joint in limbs i, ${\dot{\alpha }}_i$ , and ${\dot{\beta }}_i$ , and ${\dot{\varphi }}_{ij}$ (j = 3, 4, 5) represent angular rates of the rT joint and spherical joint in limb i.

Based on the actuation analysis in Section 5, the translation of the prismatic joint is chosen as the input for the (rT)₂PS limb and the rotation about the bracket axis is taken as the second actuation when the limb changes to phase (rT)₁PS. Thus by locking the active joints in the limbs temporarily, and taking the reciprocal product on both sides of (10), for each limb there is

${\left[\begin{array}{cc}{\mathbf{S}}_{i\mathrm{1}}^r& {\mathbf{S}}_{i\mathrm{2}}^r\end{array}\right]}^T\circ {\mathbf{S}}_G=\left\{\begin{array}{cc}{\left[\begin{array}{cc}{\dot{l}}_i& \mathrm{0}\end{array}\right]}^T\hfill & {\left(\text{rT}\right)}_{\mathrm{2}}PS\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{limb}\hfill \\ {\left[\begin{array}{cc}{\dot{l}}_i& {\dot{\alpha }}_i\end{array}\right]}^T\hfill & {\left(\text{rT}\right)}_{\mathrm{1}}PS\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{limb},\hfill \end{array}\right)$

(11)

where S _ij ^r are the reciprocal screws to the motion screws in limb i.

Equations in (11) for the three limbs can be rewritten in matrix form as

$\left[\begin{array}{c}{\mathit{S}}_{\mathrm{11}}^r\\ {\mathit{S}}_{\mathrm{21}}^r\\ {\mathit{S}}_{\mathrm{31}}^r\\ {\mathit{S}}_{\mathrm{12}}^r\\ {\mathit{S}}_{\mathrm{22}}^r\\ {\mathit{S}}_{\mathrm{32}}^r\end{array}\right]\circ {\mathit{S}}_G=\mathit{J}{\mathit{S}}_G=\left[\begin{array}{c}\begin{array}{cc}{\mathit{b}}_{\mathrm{1}}\times {\mathit{u}}_{\mathrm{1}}& {\mathit{u}}_{\mathrm{1}}\end{array}\\ \begin{array}{cc}{\mathit{b}}_{\mathrm{2}}\times {\mathit{u}}_{\mathrm{2}}& {\mathit{u}}_{\mathrm{2}}\end{array}\\ \begin{array}{cc}{\mathit{b}}_{\mathrm{3}}\times {\mathit{u}}_{\mathrm{3}}& {\mathit{u}}_{\mathrm{3}}\end{array}\\ {\mathit{j}}_{\mathrm{12}}^{}\\ {\mathit{j}}_{\mathrm{22}}^{}\\ {\mathit{j}}_{\mathrm{32}}^{}\end{array}\right]{\mathit{S}}_G=\left[\begin{array}{c}{\dot{l}}_{\mathrm{1}}\\ {\dot{l}}_{\mathrm{2}}\\ {\dot{l}}_{\mathrm{3}}\\ g_{\mathrm{1}}\\ g_{\mathrm{2}}\\ g_{\mathrm{3}}\end{array}\right],$

(12)

where

$\begin{array}{c}{\left(\text{rT}\right)}_{\mathrm{2}}PS:{\mathit{j}}_{i\mathrm{2}}^{}=\left[\begin{array}{cc}{\mathit{a}}_i\times \mathit{n}& \mathit{n}\end{array}\right],\hspace{1em}{g}_i=\mathrm{0};\\ {\left(\text{rT}\right)}_{\mathrm{1}}PS:{\mathit{j}}_{i\mathrm{2}}^{}=\left[\begin{array}{cc}{\mathit{a}}_i\times (\mathit{n}\times {\mathit{u}}_i)& \mathit{n}\times {\mathit{u}}_i\end{array}\right],\hspace{1em}{g}_i={\dot{\alpha }}_i.\end{array}$

(13)

Thus J is the Jacobian matrix. The first three rows of J represent three forces of actuation of the prismatic joints in the limbs (rT)₂PS and the last three rows are three forces from geometric constraint of the limbs (rT)₂PS or the actuated rotation of the bracket axis of the rT joints in the (rT)₁PS. Based on this, possible singular configurations can be identified for all the configurations.

In general, the Jacobian matrix maps the velocities between the manipulator and the actuation input. Once the manipulator meets the singular configuration, this mapping loses its function and the rank of the Jacobian matrix decreases to be less than 6. This can also be interpreted that the six constraint forces in J are linearly dependent. Inversely, identifying the dependent conditions for the constraint forces in the workspace will reveal the singular configurations of the manipulator. In the following, singularities are named as singularity I, II, to singularity VI.

6.1. Singularity of the 3(rT)₂PS (3DOF)

For the 3(rT)₂PS, the Jacobian matrix can be specified as

$\left[\begin{array}{c}{\mathbf{S}}_{\mathrm{11}}^r\\ {\mathbf{S}}_{\mathrm{21}}^r\\ {\mathbf{S}}_{\mathrm{31}}^r\\ {\mathbf{S}}_{\mathrm{12}}^r\\ {\mathbf{S}}_{\mathrm{22}}^r\\ {\mathbf{S}}_{\mathrm{32}}^r\end{array}\right]\circ {\mathbf{S}}_G=\left[\begin{array}{c}{\mathbf{S}}_{\mathrm{1}pa}^r\\ {\mathbf{S}}_{\mathrm{2}pa}^r\\ {\mathbf{S}}_{\mathrm{3}pa}^r\\ {\mathbf{S}}_{\mathrm{1}}^r\\ {\mathbf{S}}_{\mathrm{2}}^r\\ {\mathbf{S}}_{\mathrm{3}}^r\end{array}\right]\circ {\mathbf{S}}_G=\mathbf{J}{\mathbf{S}}_G=\left[\begin{array}{c}\begin{array}{cc}{\mathbf{b}}_{\mathrm{1}}\times {\mathbf{u}}_{\mathrm{1}}& {\mathbf{u}}_{\mathrm{1}}\end{array}\\ \begin{array}{cc}{\mathbf{b}}_{\mathrm{2}}\times {\mathbf{u}}_{\mathrm{2}}& {\mathbf{u}}_{\mathrm{2}}\end{array}\\ \begin{array}{cc}{\mathbf{b}}_{\mathrm{3}}\times {\mathbf{u}}_{\mathrm{3}}& {\mathbf{u}}_{\mathrm{3}}\end{array}\\ \begin{array}{cc}{\mathbf{a}}_{\mathrm{1}}\times \mathbf{n}& \mathbf{n}\end{array}\\ \begin{array}{cc}{\mathbf{a}}_{\mathrm{2}}\times \mathbf{n}& \mathbf{n}\end{array}\\ \begin{array}{cc}{\mathbf{a}}_{\mathrm{3}}\times \mathbf{n}& \mathbf{n}\end{array}\end{array}\right]{\mathbf{S}}_G=\left[\begin{array}{c}{\dot{l}}_{\mathrm{1}}\\ {\dot{l}}_{\mathrm{2}}\\ {\dot{l}}_{\mathrm{3}}\\ \mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array}\right],$

(14)

where S _i ^r is the geometric constraint force in limb i and S _ipa ^r is its actuation force from the prismatic joint as shown in Figure 9.

Figure 9:

Singularity I.

In Figure 9, singularity I exists when the platform plane ∑A₁A₂A₃ is parallel to the bracket axis n . In this case, the three geometric constraint forces S₁ ^r , S₂ ^r , and S₃ ^r are parallel and in the same plane ∑A₂A₁A₃. Thus, they are dependent and one of the last three rows in Jacobian J in (14) is redundant, resulting in a free rotation about any side of the platform triangle A₁A₂A₃. This was defined as Type 2b.2 singularity in [30]. Based on the geometric condition in Section 3.2, to have this singular configuration, it needs the plane ∑3 to be tangible with the constraint circle as ∑3′ or ∑3^′′. Thus, angle β₁₃ will be the maximum value β_{13max ⁡} or minimum β_{13min ⁡} in (6) which is the extreme condition to design the third limb location after fixing two of them.

Another singular configuration for the 3(rT)₂PS occurs when the three rotation angles (α₁, α₂, and α₃) about the bracket axes are equal to each other, which is defined as singularity II. In this case, the three limbs are parallel to each other. Defining limb constraint plane ∑i (i = 1, 2, 3) formed by actuation force S _ipa ^r and geometric constraint force S _i ^r as in Figure 10(a), the three limb constraint planes are parallel to each other. Thus, the six rows of J are redundant with three pairs of them distributed in three parallel planes, resulting in a free translation along the direction perpendicular to the limb constraint planes ∑i. This was taken as Type 5b.2 singularity in [30].

Figure 10:

Singularity II.

Two special cases of singularity II may exist. One is that α₁ = α₂ = α₃ = π/2 as in Figure 10(b); the platform is parallel to the base with limbs perpendicular to them, which will never happen if the base triangle (B₁B₂B₃) has a larger size than the platform (A₁A₂A₃). The other one is that α₁ = α₂ = α₃ = 0 or π as in Figure 10(c), and the platform is coincident with the base, which requires the same geometric condition for the three limbs as the singularity I configuration in Figure 9 that the third limb should be in the extreme locations shown in Section 3.2. Furthermore, these two singular configurations not only have singularity II but also include singularity I in Figure 9 as the platform plane ∑A₁A₂A₃ is parallel to the bracket axis n . Thus the rank of the Jacobian matrix J decreases to 4, leading to the fact that the platform has a free translation along the direction perpendicular to the limb constraint planes and a free rotation about any side of the platform triangle A₁A₂A₃.

6.2. Singularity of the 2(rT)₂PS-1(rT)₁PS (4DOF)

When changing to 2(rT)₂PS-1(rT)₁PS (take limb 2 to be (rT)₁PS as an example), the limb 2 constraint plane changes from ∑2 to ∑2r formed by two actuation forces S_2pa ^r from the prismatic joint and S_2ra ^r from the bracket rotation in the rT joint as in Figure 11. Both singularity I and singularity II can exist for 2(rT)₂PS-1(rT)₁PS if the geometric conditions are satisfied as investigated in Section 6.1 due to the fact that the 2(rT)₂PS-1(rT)₁PS covers the 3(rT)₂PS configuration by taking β₂ = 0. In addition to these two singularities, the 2(rT)₂PS-1(rT)₁PS parallel mechanism may also have singularity III and singularity IV as in Figure 11. Singularity III is shown in Figure 11(a), in which the common line of limb constraint planes ∑1 and ∑3 has an intersecting point Q with the common line of platform plane ∑A₁A₂A₃ and limb constraint plane ∑2r. This singularity contains more than five skew lines and was named as Type 5a singularity [30, 33]. In this case, rank of the Jacobian matrix J is 5 and the manipulator gains one free DOF as a screw motion along the direction perpendicular to the two common lines.

Figure 11:

Two singular configurations of the 2(rT)₂PS-1(rT)₁PS.

Singularity IV occurs when the limb constraint plane ∑2r is coplanar with the platform plane ∑A₁A₂A₃. In this case, all the six constraint forces intersect at one common line A₁A₃ and the Jacobian matrix rank decreases to five. The manipulator has a free rotation about the common line A₁A₃ since all the six constraint forces cannot supply any torque to this line.

6.3. Singularity of the 1(rT)₂PS-2(rT)₁PS (5DOF)

When further altering to 1(rT)₂PS-2(rT)₁PS (limb 1 is changed), the limb 1 constraint plane becomes ∑2r formed by two actuation forces as in Figure 12(a). All above singularities may exist with this topology as it covers both 2(rT)₂PS-1(rT)₁PS (β₁ = 0) and 3(rT)₂PS (β₁ = β₂ = 0). One more singularity occurs when limb constraint planes ∑1r and ∑2r intersect at line A₁A₂, named as singularity V. In this condition, actuation forces S_1ra ^r and S_2ra ^r are collinear, resulting in a redundant row in the Jacobian matrix and the manipulator has one free screw motion determined by the other five rows in J . This singularity was called Type 1 in [30, 33].

Figure 12:

Singularity V and singularity VI.

6.4. Singularity of the 3(rT)₁PS (6DOF)

When changing to the 3(rT)₁PS, the singularities in the preceding analysis will all be included once the geometric conditions are satisfied. One more singularity, number VI, will occur when all the three rotational angles β₁ = β₂ = β₃ = 0 as in Figure 12(b). In this case, the six actuation forces are distributed in three parallel limb constraint planes (∑1r, ∑2r, and ∑3r), resulting in a free rotation DOF about the direction perpendicular to the parallel planes. This is similar with singularity II and Type 5b.2 in [30].

From the above, it can be seen that when the mechanism topology changes with mobility increases, the possible singularity configurations will also increase due to the more flexible motion of the limbs. Considering the extreme limb locations for singularity I and special cases of singularity II, proper mechanism design can help avoid them. Singularity VI can also be avoided by selecting one rotation about the radial axis in one limb and other singularities need to be considered in the trajectory plan.

7. Conclusions

This paper investigated possible singularities of a class of 3rTPS metamorphic parallel mechanisms considering the geometric constraint change among the variable topologies. By changing the direction of the rotational axis of the reconfigurable Hooke (rT) joint in the rTPS limb, the limb twist system changes from order 6 to order 5 which supplies a constraint force to the platform with the direction parallel to the bracket axis and passing through the spherical joint center. Based on this, the platform constraint screw system was demonstrated to be able to change from 0 to 3, indicating that the new metamorphic parallel mechanisms have four topologies with mobility change among 1R2T, 2R2T, and 3R2T and mobility 6. Considering the kinematics solutions, geometric conditions of the mechanism design were identified and the third limb location should be inside a specific range after locating the first two limbs. Some special topologies were also listed considering special limb arrangements on the base and platform. The work was then extended to actuation scheme analysis by showing variable actuation forces. Following these, a unified Jacobian matrix was constructed by including both geometric constraint forces and actuation forces. By keeping the matrix size 6 by 6, three geometric constraint forces were changed to three actuation forces in the topology reconfiguration which gives the unified modeling to analyze possible singular configurations covering all the topologies. Six different singularities were illustrated and the general trend is that more singularities will appear with the mobility increasing due to more flexible motions in the limbs. By proper design and actuation selection, some singularities can be avoided in the mechanism configurations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.