## 1. Introduction

- (i)
- the axes of the two intermediate R-pairs are parallel to one another, and
- (ii)
- the axes of the two ending R-pairs are parallel to each other.

- (a)
- the platform and the base are manufactured so that the three R-pair axes embedded in them have a common intersection point;
- (b)
- each UPU limb (Figure 5) is manufactured and assembled so that the axes of the two intermediate R-pairs are parallel to one another; and
- (c)
- the 3-UPU is assembled so that the axes of the six R-pairs adjacent to the base or to the platform share a common intersection point (such point becomes the spherical motion center).

## 2. Background, Notations, and General Comments

- -
- Ox
_{b}y_{b}z_{b}and Px_{p}y_{p}z_{p}are two Cartesian references fixed to the base and the platform, respectively; - -
- A
_{i}(B_{i}) for i = 1, 2, 3 are the centers of the U joints adjacent to the base (platform); - -
- in each UPU limb, the four R-pairs are numbered with an index, j, that increases by moving from the base toward the platform;
- -
**w**_{ji}, for j = 1, …, 4, is the j-th R-pair axis’ unit vector of the i-th UPU limb, i = 1, 2, 3;- -
**w**_{2i}and**w**_{3i}are perpendicular to the axis of the i-th limb (i.e., the line through A_{i}and B_{i}), for i = 1, 2, 3.

- -
- θ
_{ji}, for j = 1, …,4, is the angular joint variable, counterclockwise with respect to**w**_{ji}, of the j-th R-pair of the i-th UPU limb, i = 1, 2, 3; - -
- d
_{i}= |B_{i}− A_{i}| is the linear joint variable of the P-pair (hereafter named “limb length”) of the i-th UPU limb, i = 1, 2, 3; - -
**p**= (P − O);**b**_{i}= (B_{i}− O) =**p**+**b**_{0i}with**b**_{0i}= (B_{i}− P), for i = 1, 2, 3;- -
**a**_{i}= (A_{i}− O), for i = 1, 2, 3;**c**_{i}= (**b**_{0i}−**a**_{i}) for i = 1, 2, 3;**g**_{i}= (**b**_{i}−**a**_{i})/d_{i}for i = 1, 2, 3;- -
**r**_{i}=**w**_{1i}×**w**_{2i}for i = 1, 2, 3;**h**_{i}=**w**_{3i}×**w**_{4i}for i = 1, 2, 3;**n**_{i}= [(**b**_{i}−**a**_{i})⋅**r**_{i}]**h**_{i}for i = 1, 2, 3.

**ω**is the angular velocity of the platform, and $\dot{x}$ denotes the time derivative of x. Equations (1a), (1b), and (1c) are formally the same that appeared in [34] for the 3-nSPU manipulator and, with the same algebraic manipulations reported in [34], they yield the following instantaneous input–output relationship:

**1**and

**0**are the 3 × 3 identity and null matrices, respectively; $\dot{d}={({\dot{\mathrm{d}}}_{1},{\dot{\mathrm{d}}}_{2},{\dot{\mathrm{d}}}_{3})}^{\mathrm{T}}$ is the vector collecting the P-pairs’ joint rates, which are the instantaneous inputs, and

**G**

^{T}= (

**g**

_{1},

**g**

_{2},

**g**

_{3}),

**K**

^{T}= (

**k**

_{1},

**k**

_{2},

**k**

_{3}),

**S**

^{T}= (

**s**

_{1},

**s**

_{2},

**s**

_{3}),

**J**

^{T}= (

**j**

_{1},

**j**

_{2},

**j**

_{3})

**k**

_{i}= (

**b**

_{i}−

**p**) ×

**g**

_{i},

**s**

_{i}=

**h**

_{i}×

**r**

_{i}− [

**g**

_{i}⋅(

**h**

_{i}×

**r**

_{i})]

**g**

_{i},

**j**

_{i}= (

**b**

_{i}−

**p**) ×

**s**

_{i}− [(

**b**

_{i}−

**a**

_{i})⋅

**r**

_{i}]

**h**

_{i}, i = 1, 2, 3.

**S J**]) is rank-deficient, a constraint singularity occurs.

**G K**]) is rank deficient, the platform can perform elementary motions without changing its operating mode, even though the actuated joints are locked. These three equations vary together with the associated singularity conditions if the input variables are changed. Nevertheless, if the actuated joints just control the limb lengths, the changes involve only the left-hand sides of these equations, leaving the 3 × 6 matrix [

**G K**] unchanged together with the associated singularities. For instance, with reference to Figure 2, if the actuated P-pair of the i-th limb is replaced by an actuated R-pair with an axis parallel to

**w**

_{2i}and

**w**

_{3i}, the following relationships hold

## 3. Translational 3-UTU

**w**

_{2i}= ±

**w**

_{3i}and (ii)

**w**

_{1i}= ±

**w**

_{4i}for i = 1, 2, 3, which yield

**h**

_{i}= ±

**r**

_{i},

**s**

_{i}= 0 and

**j**

_{i}= ±[(

**b**

_{i}−

**a**

_{i})⋅

**h**

_{i}]

**h**

_{i}. Consequently, the instantaneous input–output relationship (2) becomes

#### 3.1. Rotation (Constraint) Singularities

**b**

_{i}−

**a**

_{i})⋅

**h**

_{i}]4 the last three equations of system (7) become [12]

**h**

_{i}⋅

**ω**= 0 i = 1, 2, 3.

**ω**(i.e., a rotation (constraint) singularity occurs) if and only if

**h**

_{1}⋅(

**h**

_{2}×

**h**

_{3}) = 0.

**h**

_{i}, for i = 1, 2, 3, are coplanar (i.e., when all the intersections among the planes parallel to the U-joints’ cross links are parallel lines). Consequently, if the unit vectors

**w**

_{1i}(

**w**

_{4i}), for i = 1, 2, 3, are all parallel, this geometric condition is always satisfied5 and a structural rotation (constraint) singularity occurs [29,30,31].

_{b}y

_{b}z

_{b}, which represents a surface (rotation (constraint) singularity locus) in Ox

_{b}y

_{b}z

_{b}(the operational space) whose points locate the singular configurations where the platform can rotate. The deduction of this algebraic equation is as follows7:

**w**

_{12}⋅

**p**) (

**w**

_{13}⋅

**p**) [(

**w**

_{12}×

**w**

_{13}) ⋅

**p]**− (

**w**

_{12}⋅

**p**){[

**w**

_{12}× (

**c**

_{3}×

**w**

_{13})] ⋅

**p}**− (

**w**

_{13}⋅

**p**) {[(

**c**

_{2}×

**w**

_{12}) ×

**w**

_{13}] ⋅

**p}**+

+ {[(

**c**

_{2}×

**w**

_{12}) × (

**c**

_{3}×

**w**

_{13})] ⋅

**p**} + (

**w**

_{11}⋅

**p**) {[(

**w**

_{13}⋅

**p**)

**w**

_{13}− (

**w**

_{12}⋅

**p**)

**w**

_{12}− (

**c**

_{3}×

**w**

_{13}) +

**c**

_{2}×

**w**

_{12}] ×

**w**

_{11}

**}**⋅

**p**+

− (

**w**

_{11}⋅

**p**) (

**w**

_{12}⋅

**p**) (

**w**

_{13}⋅

**p**) [(

**w**

_{12}×

**w**

_{13}) ⋅

**w**

_{11}] + (

**w**

_{11}⋅

**p**) (

**w**

_{12}⋅

**p**) [

**w**

_{12}× (

**c**

_{3}×

**w**

_{13})] ⋅

**w**

_{11}+

+ (

**w**

_{11}⋅

**p**) (

**w**

_{13}⋅

**p**) [(

**c**

_{2}×

**w**

_{12}) ×

**w**

_{13}] ⋅

**w**

_{11}– (

**w**

_{11}⋅

**p**) [(

**c**

_{2}×

**w**

_{12}) × (

**c**

_{3}×

**w**

_{13})] ⋅

**w**

_{11}− {[(

**w**

_{13}⋅

**p**)

**w**

_{13}+

− (

**w**

_{12}⋅

**p**)

**w**

_{12}− (

**c**

_{3}×

**w**

_{13}) +

**c**

_{2}×

**w**

_{12}] × (

**c**

_{1}×

**w**

_{11})} ⋅

**p**+ (

**w**

_{12}⋅

**p**) (

**w**

_{13}⋅

**p**) [(

**w**

_{12}×

**w**

_{13}) ⋅ (

**c**

_{1}×

**w**

_{11})] +

− (

**w**

_{12}⋅

**p**) {[

**w**

_{12}× (

**c**

_{3}×

**w**

_{13})] ⋅ (

**c**

_{1}×

**w**

_{11})} − (

**w**

_{13}⋅

**p**) {[(

**c**

_{2}×

**w**

_{12}) ×

**w**

_{13}] ⋅ (

**c**

_{1}×

**w**

_{11})} +

+ [(

**c**

_{2}×

**w**

_{12}) × (

**c**

_{3}×

**w**

_{13})] ⋅ (

**c**

_{1}×

**w**

_{11}) = 0.

_{1}A

_{2}A

_{3}(B

_{1}B

_{2}B

_{3}) is an equilateral triangle, and the unit vectors

**w**

_{1i}(

**w**

_{4i}), i = 1, 2, 3, lie on three R-pair axes that have the center of this triangle as a common intersection. By choosing this center as origin O (P), and the triangle plane as the x

_{b}y

_{b}(x

_{p}y

_{p}) coordinate plane for Ox

_{b}y

_{b}z

_{b}(Px

_{p}y

_{p}z

_{p}), it is easy to realize that, in Equation (13a), the vectors

**c**

_{i}×

**w**

_{1i}, for i = 1, 2, 3, are all null vectors since

**c**

_{i}is parallel to

**w**

_{1i}. Consequently, when P lies on the line through O perpendicular to the base triangle A

_{1}A

_{2}A

_{3}(i.e., the SNU 3-UPU is at its home position), the dot products (

**w**

_{1i}⋅

**p**), for i = 1, 2, 3, are equal to zero and the left-hand side of Equation (13a), which becomes

**p**⋅(

**p**×

**p**), is identically equal to zero, that is, the platform can rotate.

_{1}A

_{2}A

_{3}(B

_{1}B

_{2}B

_{3}) is an equilateral triangle, but the i-th unit vector

**w**

_{1i}(

**w**

_{4i}), i = 1, 2, 3, is parallel to the base-triangle (platform-triangle) side opposite to the vertex A

_{i}(B

_{i}), which the corresponding R-pair axis passes through. For this geometry, Equation (13) yields, as a singularity locus, a cubic surface (see [14]), that is the product of the base-triangle plane by a right circular cylinder whose generatrix is a line perpendicular to the base-triangle plane. The analytic expression of this cylinder is reported in [14].

_{1}(B

_{1}) as origin O (P), and the base-triangle (platform-triangle) plane as x

_{b}y

_{b}(x

_{p}y

_{p}) coordinate plane for Ox

_{b}y

_{b}z

_{b}(Px

_{p}y

_{p}z

_{p}), the analytic expression of this cylinder is reported in [12] together with the simple geometric construction shown in Figure 3, which allows to draw immediately the singularity cylinder. The construction of Figure 3 relies on the fact that three points are sufficient to identify a circle, and that three singularities, B

_{1}’, B

_{1}’’ and B

_{1}’’’ in Figure 3, are easy to find. The same construction highlights (Figure 4) that, when any two R-pair axes (together with the corresponding unit vectors

**w**

_{1i}) are parallel, the singularity cylinder degenerates into a singularity plane (see [12] for details).

#### 3.2. Translation Singularities

**ω**, is a null vector. Thus, the first three equations of system (7) become [12]:

**g**

_{1}⋅(

**g**

_{2}×

**g**

_{3}) = 0.

**g**

_{i}, for i = 1, 2, 3, are parallel to a plane (i.e., when all the limb axes are parallel to a unique plane). If the base and platform triangles are equal, this condition is always satisfied and a structural translation singularity occurs [2,12].

_{b}y

_{b}z

_{b}, which represents a surface (translation singularity locus) in Ox

_{b}y

_{b}z

_{b}whose points locate the singular configurations where the platform translation is not controllable by the actuators. The deduction of this algebraic equation is as follows:

**p**⋅{[(

**b**

_{03}−

**a**

_{3}) × (

**b**

_{01}−

**a**

_{1})] + [(

**b**

_{01}−

**a**

_{1}) × (

**b**

_{02}−

**a**

_{2})] + [(

**b**

_{02}−

**a**

_{2}) × (

**b**

_{03}−

**a**

_{3})]} + (

**b**

_{01}−

**a**

_{1}) ⋅ [(

**b**

_{02}−

**a**

_{2}) × (

**b**

_{03}−

**a**

_{3})] = 0.

**p**in Equation (17b). By choosing A

_{1}(B

_{1}) as origin O (P), and the base-triangle (platform-triangle) plane as the x

_{b}y

_{b}(x

_{p}y

_{p}) coordinate plane for Ox

_{b}y

_{b}z

_{b}(Px

_{p}y

_{p}z

_{p}), it is easy to realize that, if the base and platform triangles are equal, the vectors (

**b**

_{0i}−

**a**

_{i}), for i = 1, 2, 3, are all null vectors. Consequently, the left-hand side of Equation (17b) is identically null (i.e., a structural singularity occurs). Also, in the case of the 3-UPUs with the axes of the three R-pairs, adjacent to the base (platform), that lie on the plane of the base (platform) triangle, it is easy to realize that the singularity plane is the base-triangle plane [12]. Indeed, in this case, the vectors (

**b**

_{0i}−

**a**

_{i}), for i = 1, 2, 3, are all parallel to the base plane, which implies that, in Equation (17b), the mixed product (

**b**

_{01}−

**a**

_{1}) ⋅ [(

**b**

_{02}−

**a**

_{2}) × (

**b**

_{03}−

**a**

_{3})] is equal to zero and the vector in curly brackets is perpendicular to the base-triangle plane.

## 4. 3-UTU Wrist

**a**

_{i}(

**b**

_{i}) parallel to

**w**

_{1i}(

**w**

_{4i}), for i = 1, 2, 3. In addition, condition (b) yields

**w**

_{2i}= ±

**w**

_{3i}; whereas, condition (c) implies that O coincides with P (i.e.,

**p**= 0). Consequently,

**k**

_{i}=

**b**

_{i}×

**g**

_{i},

**s**

_{i}=

**h**

_{i}×

**r**

_{i}since

**g**

_{i}is perpendicular to

**h**

_{i}×

**r**

_{i}, and

**j**

_{i}= 0, for i = 1, 2, 3 (see Equation (4)). These formulas allow for the conclusion that

**k**

_{i}and

**s**

_{i}are both parallel to

**w**

_{2i}and

**w**

_{3i}(see Figure 5), which are unit vectors perpendicular to the plane of the triangle A

_{i}B

_{i}P.

#### 4.1. Translation (Constraint) Singularities

**s**

_{1}⋅ (

**s**

_{2}×

**s**

_{3}) = 0,

**s**

_{i}is parallel to

**w**

_{2i}for i = 1, 2, 3, can be simplified as follows [20,21]:

**w**

_{21}⋅ (

**w**

_{22}×

**w**

_{23}) = 0.

**w**

_{2i}, for i = 1, 2, 3, are all parallel to a unique plane, that is, when the planes of the three triangles A

_{i}B

_{i}P, for i = 1, 2, 3, have a line as a common intersection (Figure 6) [20]. Also, it is worth noting that each unit vector

**w**

_{2i}is indeterminate when the triangle A

_{i}B

_{i}P is flattened (i.e., the points A

_{i}, B

_{i}, and P are aligned); in this case (see Figure 5), a simple inspection of the flattened limb reveals that the i-th limb can freely rotate around its axis.

#### 4.2. Rotation Singularities

**ω**(i.e., a rotation singularity occurs) if and only if

**k**

_{1}⋅(

**k**

_{2}×

**k**

_{3}) = 0,

**k**

_{i}is parallel to

**w**

_{2i}for i = 1, 2, 3, can be simplified as follows [20,21]:

**w**

_{21}⋅ (

**w**

_{22}×

**w**

_{23}) = 0.

## 5. Reconfigurable and Structurally Singular 3-6UTUs

**w**

_{1i}(

**w**

_{4i}), for i = 1, 2, 3, are all parallel makes the platform able to rotate around axes parallel to these unit vectors. Such additional finite DOF allows for the introduction of one more UPU limb to control the platform rotation. The resulting 4-UPU is a Shoenflies motion generator [28]. It was presented in [29] and studied in [30,31,32].

## 6. Conclusions

## Funding

## Conflicts of Interest

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1 | According to [1], here, we use the term “limb connectivity” to denote the DOF number the platform would have if it were connected to the base only through that limb. |

2 | Hereafter, U, S, R, and P stand for universal joint, spherical pair, revolute pair, and prismatic pair, respectively. Also, the underscore denotes an actuated kinematic pair. |

3 | Parallel wrists (PWs) are PMs in which the relative motion between platform and base can only be a spherical motion with a fixed center. |

4 | Since this coefficient has no effect on the value of ω when it is different from zero, the value of ω as this coefficient goes to zero is unchanged. Therefore, the zeroing of this coefficient does not affect the angular velocity of the platform and does not identify a rotation (constraint) singularity. |

5 | Indeed, in this case, all the intersections among the cross-links’ planes are lines parallel to the unit vectors w_{1i} (w_{4i}). |

6 | In TPMs, the coordinates of a platform point are sufficient to identify the platform pose in the operational space since the platform translates with respect to the base. |

7 | Note that, in a TPM, the above-defined vectors c_{i} (= b_{0i} − a_{i}), i = 1, 2, 3, are constant vectors since the platform translates. |

**Figure 1.**3-UPU architectures: (

**a**) general geometry and notations, (

**b**) 3-UPU wrist, (

**c**) Tsai 3-UPU, (

**d**) SNU 3-UPU.

**Figure 4.**Case with two parallel R-pair axes: The circular directrix of the singularity cylinder degenerates into a linear directrix (i.e., the cylinder degenerates into a plane).

**Figure 7.**DYMO 3-URU: (

**a**) 3D model, (

**b**) translational parallel manipulator (TPM) operating mode, (

**c**) parallel wrist (PW) operating mode, (

**d**) 3-DOF planar PM operating mode. Figures downloaded from http://www.parallemic.org/Reviews/Review008.html and reproduced with the permission of the authors.

**Figure 8.**Carbonari et al. reconfigurable 3-URU [26]: Joint configurations A and B refer to the TPM and PW modes, respectively, the padlock denotes the locked R-pair, the darker R-pair is the actuated pair. Figure downloaded from [36] https://www.mdpi.com/2218-6581/7/3/42 and reproduced with the permission of the authors.

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