 Original Article
 Open Access
 Published:
New Family of 3DOF UPEquivalent Parallel Mechanisms with High Rotational Capability
Chinese Journal of Mechanical Engineering volume 31, Article number: 12 (2018)
Abstract
Parallel mechanisms (PMs) having the same motion characteristic with a UP kinematic chain (U denotes a universal joint, and P denotes a prismatic joint) are called UPequivalent PMs. They can be used in many applications, such as machining and milling. However, the existing UPequivalent PMs suffer from the disadvantages of strict assembly requirements and limited rotational capability. Type synthesis of UPequivalent PMs with high rotational capability is presented. The special 2R1T motion is briefly discussed and the fact that the parallel module of the Exechon robot is not a UPequivalent PM is disclosed. Using the Lie group theory, the kinematic bonds of limb chains and their mechanical generators are presented. Structural conditions for constructing such UPequivalent PMs are proposed, which results in numerous new architectures of UPequivalent PMs. The high rotational capability of the synthesized mechanisms is illustrated by an example. The advantages of no strict assembly requirements and high rotational capability of the newly developed PMs will facilitate their applications in the manufacturing industry.
Introduction
Among the various lower mobility parallel mechanisms(PMs), those with two rotational and one translational motion(2R1T) are particularly useful and can be implemented in many applications. The Z3 head [1], whose parallel part is a 2R1T 3PRS PM(P is a prismatic pair, R is a revolute pair, and S is a spherical joint), is the most wellknown. Another example is the 2UPRSPR PM(U denotes a universal joint), which has been successfully used in the development of the Exechon robot [2].
It should be noted that different 2R1T PMs may have distinct motion characteristics. According to the virtual chain method [3], they can be classified into three categories: PUequivalent PMs, RPRequivalent PMs, and UPequivalent PMs.
A PUequivalent PM can perform three degrees of freedom (3DOF) motion that is identical to the end motion of a PU serial chain. The two axes of rotation always intersect and remain perpendicular. In other words, the rotation center of a PUequivalent PM is restricted to a line; thus, the socalled parasitic motion [4] is eliminated. There were few PUequivalent PMs until the authors of this paper systematically investigated their type synthesis [5].
An RPRequivalent PM can perform 3DOF motion that is the same as the end motion of an RPR serial chain. The two rotational axes are in different planes and are perpendicular. Li and Hervé [6, 7] disclosed numerous architectures of the RPRequivalent PMs. This type of mechanism can be used in many applications where high stiffness and dexterity are required, such as in fiveaxis machining [8], aircraft wing assembly [9], and friction stir welding [10].
A UPequivalent PM can perform 3DOF motion that is identical to the end motion of a UP serial chain. One axis of rotation is fixed to the base while intersecting and remaining perpendicular to the other. Such features lead to reduced complexity in kinematic model, control, calibration and are thus preferable in practice. There are two ways to construct UPequivalent PMs. The first is to either add a passive UP serial chain to a 6DOF 3UPS PM, like the Tricept robot [11], or add an articulated UP serial chain to a 6DOF 2UPS PM, like the TriVariant robot [12, 13]. The passive or articulated UP serial chain constrains the motion of the moving platform, and no strict geometrical condition is required. Thus, the Tricept and TriVariant robots are widely used in manufacturing and assembly.
The second is to use identical limb chains. Kong and Gosselin first studied the type synthesis of UPequivalent PMs using screw theory [3]. They systematically enumerated numerous architectures. However, these architectures require strict geometrical conditions, such as the intersection of three or more revolute axes at a common point in space, which is very difficult to implement in practice.
In addition, one may think that the parallel module of the Exechon robot [2], which is a 2UPRSPR PM, belongs to the UPequivalent PMs family. In the following section, we demonstrate that the mechanism is not a UPequivalent PM.
Although the UPequivalent PMs output 2R1T motion, their reachable workspaces are usually of concern since they are used for locating in most applications. However, it should be noted that their reachable workspaces are highly dependent on their rotational capabilities since the position variations of the moving platforms in two directions are generated by the two rotational motions of the mechanisms. The reachable workspaces of the Tricept and TriVariant robots are not large because their rotational capabilities are limited to some extent by spherical joints, particularly with the existence of multiple spherical joints. It is meaningful to expand the rotational capabilities of UPequivalent PMs to ultimately achieve considerable reachable workspaces.
In this paper, we attempt to invent new architectures of UPequivalent PMs with high rotational capability and low geometrical condition requirements. That is, the new architectures of UPequivalent PMs should contain no spherical joints and should avoid a common point of intersection of several revolute axes in space. Theoretically, type synthesis of PMs with two rotational DOFs is more difficult and complicated than that of PMs with either one or no rotation. Hence, although considerable progress has been made in terms of the general method for type synthesis of lowermobility PMs [14,15,16,17,18,19,20,21,22,23,24,25,26,27], and some 3T or 3T1R PMs with specified kinematic properties have been proposed [28,29,30,31,32], few architectures of UPequivalent PMs without spherical joints and strict assembly requirement have been disclosed.
The organization of this paper is as follows. Section 2 presents a brief introduction of notations and properties of the UPequivalent motion. Section 3 discusses the limb bonds of the UPequivalent PMs. Section 4 constructs the UPequivalent PMs in a special category. Section 5 illustrates the rotational capability of the proposed mechanisms. Section 6 discusses the application of the proposed PMs. Section 7 summarizes the conclusions.
Special 2R1T Motion
Notations
Firstly, a brief introduction of the Lie group theory is given. Some displacement subgroups used in this paper are listed in Table 1. For more details about Lie group theory, please see Refs. [14,15,16,17,18,19,20].
Since there are usually some special geometrical conditions in lower mobility PMs, superscripts are used in the following sections to indicate the axial direction of the kinematic pairs. For instance, ^{u}P represents a prismatic pair with its axis parallel to vector u; ^{u}R denotes a revolute pair whose axis is parallel to vector u; and ^{uv}U represents a universal joint whose two rotational axes are parallel to vectors u and v, respectively. In contrast, the spherical joint is denoted by S without superscript since it allows rotation around any axis passing through its center.
In view of practical values, limbs that contain two or more prismatic pairs, helical pairs, and parallelogram are neglected here. However, UPequivalent PMs with these pairs can also be synthesized following the method raised in this paper.
Special 2R1T Motion
A special 2R1T motion can be represented by product {R(O, u)}{R(O, v)}{T(w)}. Such a motion can be generated by a ^{uv}U^{w}P chain or a ^{u}R^{v}R^{w}P chain. The rotation center O is fixed, and thus no parasitic motion exists.
A displacement set of {R(O, u)}{R(O, v)}{T(w)} is obtained by the composition of three displacements. The first displacement is a translation parallel to w:
where the real number t is the translation amplitude.
The second displacement is a rotation of angle ψ around axis (O, v):
Since OM_{T} = tw + OM, the transformation of M into M_{R1} is expressed by
The third displacement is a rotation of angle φ around axis (O, u). The point M_{R1} is transformed into the final position M′ as
which expresses the transformation M→M′.
The displacement M→M′ is the product of spherical rotation M→M_{ R } = O+exp(φu×)exp(ψv×)(OM), followed by translation M_{ R }→M′ = M_{ R } + texp(φu×)exp(ψv×)w.
Special Case: Exechon Robot
Figure 1 shows a 2UPRSPR PM that is the parallel module of the Exechon robot. Intuitively, it is often thought that the 2UPRSPR PM is a UPequivalent PM. However, this is not true.
Limb 1 and limb 3 are UPR chains. Limb 2 is an SPR chain. A_{1} and A_{3} denote the central points of the two Ujoints in limb 1 and limb 3, respectively. A_{2} denotes the center of the spherical joint in limb 2. B_{ i }(i = 1, 2, 3) denotes a point on the revolute axes adjacent to the moving platform in limb i. Counting from the base, the first revolute axes of the two Ujoints are coincident. The fourth revolute axes in limb 1 and limb 3 are parallel, and perpendicular to the fifth revolute axis in limb 2.
The kinematic bond of limb 1 is {L_{1}} = {R(A_{1}, u)}{R(A_{1}, v)}{T(x_{ 1 })}{R(B_{1}, v)}, where x_{1} is a unit vector in the uw plane and is perpendicular to v. Using product closure, we have {L_{1}} = {R(A_{1}, u)}{G(v)} = {R(O, u)}{G(v)}, because O is also a point on the first revolute axis.
The kinematic bond of limb 3 is {L_{3}} = {R(A_{3}, u)}{R(A_{3}, v)}{T(x_{ 3 })}{R(B_{3}, v)}, where x_{3} is a unit vector in the uw plane and is perpendicular to v. Through a similar analysis to that of limb 1, we have {L_{3}} = {R(O, u)}{G(v)}.
Since {L_{1}} = {L_{3}}, we only need to discuss the intersection of the displacement sets produced by limb 1 and limb 2. The kinematic bond of limb 2 is {L_{2}} = {R(A_{2}, v)}{R(A_{2}, u)}{R(A_{2}, x_{ 2 })}{T(x_{ 2 })}{R(B_{2}, u)}, where x_{2} is a unit vector in the vw plane and is perpendicular to u.
Because {R(A_{2}, v)}{R(A_{2}, u)}{R(A_{2}, x_{ 2 })} = {S(A_{2})}, we have
The intersection of limb 1 and limb 2 is given by
It is simple to determine {G(v)}∩{G(u)} = {T(w)}. Hence, {G(v)}and {G(u)}can be decomposed into products, including the factor {T(w)}, as
Substituting Eq. (7) and Eq. (8) into Eq. (6) yields
Because {R(O, u)}{R(O, v)}≠{R(O, v)}{R(O, u)}, {R(O, u)}{R(O, v)}{T(u)}∩{S(A_{2})}{G(u)} is only a twodimensional submanifold of {R(O, u)}{R(O, v)}{T(u)}. Hence, although the parallel module of the Exechon robot has three DOFs, it is not a UPequivalent PM.
Limb Bond of UPEquivalent PMs
Displacement Set of UPEquivalent PMs
The motion generated by a UP chain has three DOF. Hence, three limbs are preferable for the UPequivalent PMs. Each limb generates a set of feasible displacements (or kinematic bond) denoted by {L_{ i }}, (i = 1, 2, 3). {R(O, u)}{R(O, v)}{T(w)} must be the intersection of all limb kinematic bonds produced by all limb chains. Hence, we can write
Limb Bond of UPEquivalent PMs
Obviously, {R(O, u)}{R(O, v)}{T(w)} should be included in the displacement set {L_{ i }} generated by the ith limb chain, that is, {R(O, u)}{R(O, v)}{T(w)}⊂{L_{ i }}. If one or two limbs are sufficient to produce a UP motion, one or two generators of {D} can be the second and third limb. For example, one can construct a UPequivalent PM using two 6DOF SPS or UPS limbs, and one ^{uv}U^{w}P chain. This is straightforward and is not discussed in this paper. We only focus on the type synthesis of UPequivalent PMs with lowermobility limbs.
There are three groups of limb bonds, distinguished by their dimensions (Dim).
 Group 1::

Dim({L_{ i }}) = 3
This corresponds to only one condition, i.e., {L_{ i }} = {R(O, u)}{R(O, v)}{T(w)}. The corresponding mechanical generator (MG) is a 3DOF ^{uv}U^{w}P chain. However, one cannot construct a UPequivalent PM with three identical ^{uv}U^{w}P chains.
 Group 2::

Dim({L_{ i }}) = 4
To obtain 4DOF limbs, one can add {R(O, u)}{R(O, v)}{T(w)} to a 1D subgroup. Only results that yield subgroups of {D} are considered here. A 1D translational subgroup or rotational subgroup can be selected as the added factor.
Adding {R(O, u)}{R(O, v)}{T(w)} to a 1D translational subgroup {T(t)} yields
{T2(w, t)} is generated by a serial array of two prismatic pairs. However, it is discarded because limbs with two prismatic pairs are not considered for practical purposes.
With added factor {R(A, w)}, we have
Note that the point A does not necessarily belong to axis (O, w). However, if the endeffector motion is constrained by the other two limbs to be {R(O, u)}{R(O, v)}{T(w)}, then there is no motion in the R pair embodying {R(A, w)}. Hence, combination {R(O, u)}{R(O, v)}{C(A, w)} is discarded.
If A∈axis(O, w), Eq. (12) becomes
The 3D group {S(O)} is equal to product{R(O, i)}{R(O, j)}{R(O, k)} provided that (i, j, k) is a vector base. With k = w, we have{R(O, i)}{R(O, j)}{R(O, w)}{T(w)} = {R(O, i)}{R(O, j)}{C(O, w)} and we can use {C(O, w)} = {T(w)}{R(O, w)}, which gives {R(O, i)}{R(O, j)}{T(w)}{R(O, w)}, where i≠u and j≠v.
With added factor {R(A, v)}, we have
The 3D group {G(v)} is equal to {R(A, v)}{R(B, v)}{T(r)} with generally A≠O and r⊥v≠w. We also have {G(v)} = {R(A, v)}{T(r)}{R(B, v)} = {T(r)}{R(A, v)}{R(B, v)} = {R(A, v)}{R(B, v)}{R(C, v)}.
In summary, we obtain two types of 4D products: {S(O)}{T(w)} = {R(O, i)}{R(O, j)}{C(O, w)} with i ≠ u and j≠v, and {R(O, u)}{G(v)}, in which the cylindrical 2D group, spherical 3D group, and planar 3D group can be decomposed into products of their 1D subgroups. Table 2 lists the 4D limb bonds and their MGs of UPequivalents PMs.
 Group 3::

Dim({L_{ i }}) = 5
With added factor {R(A, k)}{R(B, k)}, we have
Thus, various MGs can be obtained by using product closure in {G(k)}, as listed in Table 3.
Adding a group of translation {T(t)}, and ∀ unit vector t ≠ w, in the 4D product {S(O)}{T(w)}, we obtain
{S(O)}{T2(w, t)} is an irreducible representation of the reducible product {S(O)}{G(s)}, with s⊥w (the planes orthogonal to s are parallel to w).
{S(O)}{G(s)} has two main categories of irreducible representations. One is {R(O, i)}{R(O, j)}{G(s)}, where (i, j, k) is a vector base (generally, i ≠ u and j ≠ v). The other is {S(O)}{G2(s)}, where {G2(s)} denotes one of the 2D submanifolds of the 3D group{G(s)}. We neglect {S(O)}{G2(s)} intentionally because a spherical joint is needed to implement {S(O)}. Table 4 lists the MGs of 5D limb bond {R(O, i)}{R(O, j)}{G(s)}.
Adding a 2D group of planar translations in product {R(O, u)}{R(O, v)}{T(w)}, we obtain
{X(u)}{X(v)} is called the double Schoenflies motion or X–X motion [18]. It is reducible and includes three translations and two rotations. The axes of the rotations are parallel to vectors u and v, whereas the locations of the axes can be arbitrary. For detailed MGs of the X–X motion, please see Ref. [18].
Parallel Arrangement of Limbs
A UPequivalent PM can be obtained by connecting a moving platform to a fixed base using the three limbs listed in Table 2 or Table 3. Note that the Ujoints in the three limbs must be coincident, which causes difficulties in fabrication and actuation. Henceforth, we use the first revolute axis to denote the fixed revolute axis of the Ujoint and the second axis to denote the moving revolute axis of the Ujoint.
We set the first revolute axes of the Ujoints in limbs 2 and 3 to be coincident with the first revolute axis of the Ujoint in limb 1, as shown in Figure 2(a). Consequently, we have {R(O, u)}∩{R(A_{2}, u)}∩{R(A_{3}, u)} = {R(O, u)} because the three Ujoint centers O, A_{2}, and A_{3} are collinear.
Then, the two Ujoints in limb 1 and limb 3 are combined into one complex Ujoint as shown in Figure 2(b). Note that this condition implies that {L_{1}} = {L_{3}} and the three limbs are set orthogonally. Consequently, the displacement set of the moving platform is determined by the intersection of {L_{1}} and {L_{2}}.
For clarity, we denote the type of UPequivalent PMs by the dimensions of the three limbs, that is, Dim({L_{1}})Dim({L_{2}})Dim({L_{3}}). In this way, we classify the UPequivalent PMs into three categories: 33X, 44X, and 55X, in which X can be 3, 4, or 5. Here we only focus on subcategory 554 for practical purposes.
UPEquivalent PMs in Category554
There are three limb bonds for a 55X UPequivalent PM, namely, {R(O, u)}{R(O, v)}{G(k)}, {G(u)}{G(v)}, and {S(O)}{G(v)}. One can set any of them to be {L_{1}} and one of the other two to be {L_{2}}. However, for practical reasons, we let {L_{1}} = {L_{3}} = {R(O, u)}{R(O, v)}{G(k)}.
When {L_{2}} = {R(O, u)}{G(v)}, the intersection of {L_{1}} and {L_{ 2 }} is {R(O, u)}{R(O, v)}{T(w)}. Using the four MGs of {R(O, u)}{R(O, v)}{G(k)} and the four MGs of {R(O, u)}{G(v)}, one can obtain 16 UPequivalent PMs, as listed in Table 5. Figure 3 shows three UPequivalent PMs, in which the first three kinematic pairs, counting from the moving platform in each of the three architectures, are identical; this results in a maximum symmetry in the UPequivalent PMs. Figure 4 shows another six UPequivalent PMs in this group.
Note that {S(A_{ 2 })}{T(w)} cannot be used as a limb bond in this group. When {L_{2}} = {S(A_{ 2 })}{T(w)}, the intersection of {L_{1}} and {L_{ 2 }} is
Another option is to let {L_{1}} = {L_{3}} = {G(u)}{G(v)}. When {L_{2}} = {R(O, u)}{G(v)}, the intersection of {L_{1}} and {L_{ 2 }} is {L_{ 2 }} itself. Thus, {R(O, u)}{G(v)} cannot be used as a limb bond in this group.
When {L_{2}} = {S(A_{2})}{T(w)} and A_{2}∈axis (O, u), the intersection of {L_{1}} and {L_{ 2 }} is
Using the fifteen MGs of {G(u)}{G(v)} and the three MGs of {S(A_{ 2 })}{T(w)}, one can obtain 45 UPequivalent PMs. However, there is less practical potentials of those architectures due to the difficulties in actuation.
Rotational Capability
It is obvious that the constructed PMs have unlimited rotational capability around one axis, i.e., the coincident axis of the two Ujoints in each mechanism. In contrast, the rotational range around the other axis can also be considerably large, which is proved in this section by taking the 2UPRR/UPR mechanism as an example. As shown in Figure 5, the mechanism has three limbs; limb 1 and limb 2 are UPRR type limbs and share a common Ujoint, and limb 3 is a UPR type limb. The central points of the P pair and the first R pair in limb 1 (limb 2) are coincident, and these points are denoted by A_{1} (for limb 1) and A_{2} (for limb 2) hereafter. The center of the Ujoint in limb 3 is denoted by A_{3}, and the centers of the three moving platformconnected R pairs are denoted by B_{1}, B_{2}, and B_{3}, respectively.
There are several necessary geometrical conditions in the mechanism. The first rotational axes of the two Ujoints (s_{1}, s_{7}) are coincident. In limb 1 and limb 2, the R pairs have parallel axes (s_{3}, s_{4}, s_{5}, s_{6}). In limb 3, the R pair has an axis (s_{9}) parallel to line B_{1}B_{2} and the second rotational axis (s_{8}) of the Ujoint within the same limb. The three P pairs (denoted by P_{ i } for limb i) are equipped with motors. In addition, the rotation around axis s_{1} is also actuated. Thus, there are four inputs in total. In an initial configuration, as shown in Figure 5(a), s_{1} and s_{7} are parallel to s_{3}, s_{4}, s_{5} and s_{6}. s_{8} and s_{9} are parallel to line B_{1}B_{2} and the second rotational axis (s_{2}) of the Ujoint in limb 1. From this configuration, if we fix P_{3} as well as axis s_{1} of the Ujoint and give P_{1} and P_{2} with different displacements, the mechanism will move to a configuration as in Figure 5(b), in which s_{2} is no longer parallel to s_{8}, s_{9}, and line B_{1}B_{2}. The mechanism degenerates into a 2DOF mechanism with one rotational DOF around axis s_{1}(s_{7}) and a translational DOF. This is denoted by 1R1T configuration. In contrast, if we fix axis s_{1}, P_{1}, and P_{2} in the initial configuration, and give a displacement to P_{3}, the mechanism will move to a configuration in Figure 5(c), in which s_{1} and s_{7} are not parallel to s_{3}, s_{4}, s_{5} and s_{6}. The mechanism in this configuration has three DOFs, including two rotations around axes s_{1} and s_{2}; this is called the 2R1T configuration. The rotational angle of s_{1} always equals that of s_{7}, and the displacements of P_{1} and P_{2} are the same since limb 1 and limb 2 are designed to have identical dimensions. Therefore, if the mechanism is in the initial configuration, to prevent it from moving to the 1R1T configuration, P_{1} and P_{2} should be properly controlled to ensure that \(\left {OA_{1} } \right = \left {OA_{2} } \right\) (O is the center of the Ujoint in limb 1). Thus, s_{2} will remain parallel to s_{8}, s_{9}, and line B_{1}B_{2}. The mechanism continues working in the 2R1T operation mode.
Next, the kinematic model of the mechanism is established. Here, we only focus on the 2R1T operation mode. A global coordinate system G(OXYZ) is attached to the fixed base with its origin coincident with the central point of the Ujoint in limb 1; axis Y is coincident with the first rotational axis of the Ujoint, which is connected to the fixed base; axis X is parallel to the horizontal plane and perpendicular to axis Y, and axis Z is defined according to the right hand rule, as in Figure 6. Similarly, a local coordinate system L(pxyz) is attached to the moving platform with its origin located at the middle of line B_{1}B_{2}, axis x coincident with B_{2}B_{1}, axis y passing through point B_{3}, and axis z defined according to the right hand rule.
The orientation of the moving platform can be expressed by a rotation around axis Y followed by a rotation around the resulting axis X’. Thus, the rotational matrix is given by
where \(\alpha\) and \(\beta\) are rotational angles around axes Y and X′, respectively.
The position vector of point p expressed in the global coordinate system can be given by
where t denotes the distance between points O and p.
Inverse Position Solutions and Jacobian Matrix
The displacements of the three P pairs are denoted by d_{1}, d_{2}, and d_{3}, and the rotational angle of the first axis of the Ujoint in limbs 1 and 2 is denoted by \(\theta\), as in Figure 6. The inverse position problem is to find the values of d_{1}, d_{2}, d_{3}, and \(\theta\) with the moving platform parameters \(\alpha\), \(\beta\), and t given. This is quite an easy problem and one can directly determine that \(\theta\) equals \(\alpha\). We express the position vectors of points A_{ i } (i = 1, 2, 3) in the global coordinate system as
where a represents the length of OA_{3}. We have d_{1} = −d_{2} since P_{1} and P_{2} are properly actuated.
The position vectors of points B_{ i } (i = 1, 2, 3) expressed in local coordinate system are given by
where the left superscript L denotes that the coordinates are expressed in the local system, m and n are dimensional parameters of the moving platform.
The vectors in Eq. (23) can be transformed into the global coordinate system using the following expression
Thus, we have
Three constraint equations are established as
where the constant l represents the length of link A_{1}B_{1} and A_{2}B_{2}.
Substituting Eqs. (22) and (25) into Eq. (26) yields
Through Eq. (27), the input parameters can be easily solved given the output parameters α, β, and t.
Differentiating both sides of Eq. (27) with respect to time, and using the fact that \(\dot{\theta }{ = }\dot{\alpha }\), a set of equations can be derived and rearranged into matrix form as
in which \(\dot{\varvec{d}}\) = [\(\dot{d}_{1}\), \(\dot{d}_{2}\), \(\dot{d}_{3}\), \(\dot{\theta }\)]^{T} is a vector that represents the input joint velocities, \(\dot{\varvec{x}}\) = [\(\dot{\alpha }\), \(\dot{\beta }\), \(\dot{t}\)]^{T} represents the velocities of the moving platform, and J is a \(4 \times 3\) matrix as follows:
in which the detailed expressions of the elements are as
Workspace and Singularity
Once the inverse kinematic model is established, the workspace of the mechanism can be determined by considering some constraints imposed by joints and actuators. Since the mechanism has unlimited rotational capability around axis Y, \(\alpha\) is fixed as zero without loss of generality, and only the ranges of \(\beta\) and t are evaluated. Table 6 lists the dimensional parameters of the mechanism. The constraints of workspace are given as
where \(\psi\) is the angle between A_{3}B_{3} and axis Y, as shown in Figure 6. \(\psi\) is constrained in a reasonable region to avoid interference between links.
By searching the potential region for points that satisfy the conditions in Eq. (30), the workspace of the mechanism is numerically determined as in Figure 7(a). The range of t is determined by the stroke of the actuators mounted to P_{1} and P_{2}. When P_{1} and P_{2} reach their maximum stroke, t has a maximum value of −216 mm, corresponding to the right boundary of the workspace. Figure 7(e) shows such a configuration. When P_{1} and P_{2} reach their minimum stroke, t has a minimum value of −349.08 mm, corresponding to the left boundary of the workspace. Such a configuration is shown in Figure 7(d). The upper and lower boundaries of the workspace are determined by the stroke of P_{3} and the angle constraint of \(\psi\).
From Figure 7(a), we can see that rotational angle \(\beta\) is allowed over a considerably large range, varying from −33° to 70°. Furthermore, when t is set to a constant value, the rotational capability is still very high. The ranges of \(\beta\) are −5.82°~63.89° and −25.85°~54.84° when t is fixed at −320 mm and −240 mm, respectively, as illustrated in Figure 7(a), Figure 7(f), and Figure 7(g).
A main factor that limits the rotational capability of a parallel mechanism is the inherent singularity, which may appear in the workspace, and divide it into small pieces. Therefore, the singularities of the 2UPRR/UPR mechanism should be considered. Singularities of a mechanism can be determined by judging the determinant of its Jacobian matrix [33]. For the redundantlyactuated mechanism studied here, the Jacobian matrix is a \(4 \times 3\) matrix, but we can easily find that J_{1} = −J_{2} (J_{1} and J_{2} are the first and second rows of the Jacobian matrix J, respectively) since d_{1} = −d_{2} always holds. Therefore, the singularities of the 2UPRR/UPR mechanism can be defined by the following expression
Using this expression, the singularities of the mechanism are found and displayed with the orange line in Figure 7(a). We find that the singularities are not within the workspace, which means the high rotational capability of the mechanism will not be affected.
In fact, the determined singularities correspond to configurations when points O, A_{3}, and B_{3} locate at a common line. Two singular configurations are illustrated in Figure 7(b) and Figure 7(c). In these configurations, the moving platform can have an infinitesimal translation in the direction perpendicular to OA_{3}, despite all the actuations being locked. In other words, any external force in this direction cannot be balanced by the actuate force/torque. However, these singular configurations are not contained in the workspace since d_{3} and \(\psi\) exceed their permitted ranges.
Application
Parallel mechanisms have been successfully applied in the development of various machine tools [34,35,36,37]. It is believed that some of the proposed UPequivalent PMs, which have compact structures and less joints in the limbs, have promising applications in this field. For example, by combining the 2URPR/UPR PM and a 2DOF translational gantry, a hybrid milling machine is developed as in Figure 8. The machine has a tool mounted on the moving platform of the PM and can operate on complex surfaces by relying on the coordinated motions of the PM and the gantry. It has the same operational ability as the Tricept and TriVariant robots. Furthermore, its parallel part has the advantages of high load capacity, high rotational capability, and no strict assembly requirement, which is preferable in practice.
Conclusions

1.
A new family of UPequivalent PMs with high rotational ability and low requirements in terms of geometrical conditions is disclosed using displacement subgroup theory. The new architectures of UPequivalent PMs do not contain spherical joints and have no strict assembly requirements; this has several benefits in terms of manufacturing cost and payload capability.

2.
The high rotational capability of the proposed mechanisms is verified by investigating a typical example. In contrast to the unlimited rotational capability around one axis, rotation around the other axis is allowed over a considerably wide range of −33° ~ 70°. The singular curve is not within the workspace, and thus does not affect the rotational capability of the mechanism.
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Authors’ contributions
QCL and WY designed and conducted the research. XXC finished 3D models of the mechanisms. All authors contributed to the preparation of the manuscript. All authors read and approved the final manuscript.
Authors’ Information
Wei Ye, born in 1988, is currently a lecturer at Zhejiang SciTech University, China. He received his Ph.D. degree on mechanism design and theory from Beijing Jiaotong University, China, in 2016. His research interests include design and analysis of reconfigurable parallel mechanisms. Email: wye@zstu.edu.cn
QinChuan Li, born in 1975, is currently a professor at Zhejiang SciTech University, China. He received his Ph.D. degree on mechanism design and theory from Yanshan University, China, in 2003. His research interests include mechanism theory of parallel manipulators and application. Tel: +8657186843686; Email: lqchuan@zstu.edu.cn.
XinXue Chai, born in 1988, is currently a lecturer at Zhejiang SciTech University, China. She received her Ph.D. degree on mechanism design and theory from Zhejiang SciTech University, China, in 2017. Her research interests include mechanism theory of parallel manipulators and application. Email: jxcxx88@163.com.
Acknowledgements
Supported by National Natural Science Foundation of China (Grant Nos. 51525504, 51475431), Zhejiang Provincial Natural Science Foundation of China (Grant No. LZ14E050005), and Science Foundation of Zhejiang SciTech University, China (Grant No. 16022091Y).
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Ye, W., Li, Q. & Chai, X. New Family of 3DOF UPEquivalent Parallel Mechanisms with High Rotational Capability. Chin. J. Mech. Eng. 31, 12 (2018). https://doi.org/10.1186/s1003301802011
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Keywords
 Parallel mechanism
 Group theory
 Type synthesis
 High rotational capability