# A Novel 3-URU Architecture with Actuators on the Base: Kinematics and Singularity Analysis

## Abstract

**:**

## 1. Introduction

- (i)
- the actuators are on the base even though the actuated joints are not on the base,
- (ii)
- in each URU limb, the actuated R-pair is the one not adjacent to the base in the U-joint adjacent to the base, and
- (iii)
- it has a particular base (platform) geometry where the axes of the three R-pairs adjacent to the base (to the platform) share a common intersection point but are not coplanar.

## 2. The Novel Translational 3-URU

- Ox
_{b}y_{b}z_{b}and Px_{p}y_{p}z_{p}are two Cartesian references fixed to the base and to the platform, respectively; without losing generality, these two references have been chosen with the homologous coordinate axes that are parallel to one another2; - A
_{i}(B_{i}) for i = 1,2,3 are the centers of the U joints adjacent to the base (to the platform); - without losing generality [15], in the i-th limb, i=1,2,3, the points A
_{i}and B_{i}are assumed to lie on the same plane perpendicular to the axes of the three intermediate R-pairs; such plane intersects at C_{i}the axis of the R-pair between the two U-joints; **e**_{1},**e**_{2}, and**e**_{3}are unit vectors of the coordinate axes x_{b}, y_{b}, and z_{b}(x_{p}, y_{p}, and z_{p}), respectively, and, at the same time, unit vectors of the three R-pair axes fixed to the base (to the platform);**g**_{i}, i = 1, 2, 3, is the unit vector parallel to the axes of the three intermediate R-pairs of the i-th limb.

- d
_{p}= B_{1}P = B_{2}P = B_{3}P; - d
_{b}= A_{1}O = A_{2}O = A_{3}O; - in each URU limb, the five R-pairs are numbered with an index, j, that increases by moving from the base toward the platform; the actuated joint is the second R-pair;
**g**_{1}= cosθ_{11}**e**_{2}+ sinθ_{11}**e**_{3},**g**_{2}= −cosθ_{21}**e**_{1}+ sinθ_{21}**e**_{3},**g**_{3}= cosθ_{31}**e**_{1}+ sinθ_{31}**e**_{2};- θ
_{iM}, for i = 1, 2, 3, is the rotation angle of the motor shaft (see Figure 2) of the actuator of the i-th limb; - f
_{i}= A_{i}C_{i}, for i = 1, 2, 3; r_{i}= B_{i}C_{i}, for i = 1, 2, 3; **h**_{i}=**g**_{i}×**e**_{i}, for i = 1, 2, 3;**u**_{i}= (C_{i}− A_{i})/f_{i}= cosθ_{i2}**e**_{i}+ sinθ_{i2}**h**_{i}, for i =1, 2, 3;**v**_{i}= (B_{i}− C_{i})/r_{i}= cosθ_{i3}**u**_{i}+ sinθ_{i3}(cosθ_{i2}**h**_{i}− sinθ_{i2}**e**_{i}) for i = 1, 2, 3, which also defines the phase reference of the angle θ_{i3};**p**= (P − O) = x**e**_{1}+ y**e**_{2}+ z**e**_{3}, where (x, y, z)^{T}collects the coordinates of point P in Ox_{b}y_{b}z_{b}; such coordinates also identify the platform pose during motion since the studied 3-URU is translational;**a**_{i}= (A_{i}− O) = d_{b}**e**_{i}, for i = 1, 2, 3;**b**_{i}= (B_{i}− O) =**p**+ d_{p}**e**_{i}, for i = 1, 2, 3;**c**_{i}= (C_{i}− O) =**a**_{i}+ f_{i}**u**_{i}, for i = 1, 2, 3.

## 3. Mobility Analysis

**ω**is the angular velocity of the platform.

**h**

_{i}(=

**g**

_{i}×

**e**

_{i}) yields

**h**

_{i}⋅

**ω**= 0 i = 1,2,3

**b**

_{i}−

**c**

_{i}) = r

_{i}

**v**

_{i}yields

**0**

_{3 × 3}is the 3 × 3 null matrix,

^{i}

**ω**

_{pq}denotes the angular velocity of link p with respect to link q in the i-th limb, and the index M denotes the motor shaft. In addition, the relative motion theorems [16] states that

_{i}be the speed ratio of the bevel gearbox of the i-th limb, the following relationship must hold:

_{iM|0}and θ

_{i1|0}are the values of θ

_{iM}and θ

_{i1}, respectively, when θ

_{i2}is equal to zero.

**g**

_{i}, after some algebraic manipulations, relates the joint rates ${\dot{\mathsf{\theta}}}_{\mathrm{i}1}$, for i = 1, 2, 3, to the platform twist as follows:

#### 3.1. Singularity Analysis

#### 3.1.1. Rotation (Constraint) Singularities of LaMaViP 3-URU

**ω**, equal to zero. The last three equations of system (5) are able to impose

**ω**= 0, if the determinant of the coefficient matrix,

**H**, is different from zero. Therefore, the constraint singularities are the configurations that satisfy the geometric condition6

**H**) =

**h**

_{1}⋅(

**h**

_{2}×

**h**

_{3}) = 0

**h**

_{i}, for i = 1, 2, 3, are coplanar. Since the i-th unit vector

**h**

_{i}is perpendicular to the plane passing through the coordinate axis of Ox

_{b}y

_{b}z

_{b}with the direction of

**e**

_{i}where the unit vector

**g**

_{i}lies on (that is, to the plane where the cross link of the i-th U-joint lies on (see Figure 1)) and the three so-identified planes always share point O as common intersection, such a geometric condition occurs when these three planes simultaneously intersect themselves in a common line passing through point O (see Figure 4).

**p**(i.e.,

**p**= x

**e**

_{1}+ y

**e**

_{2}+ z

**e**

_{3}) into Equation (15b) yields

- when point P lies on the y
_{b}z_{b}coordinate plane (i.e., x = 0), the three unit vectors**h**_{i}, for i = 1, 2, 3, (see Formulas (16)) are all parallel to the y_{b}z_{b}coordinate plane; therefore, the component of**ω**along**e**_{1}is not locked (see Equations (2)) and the platform can perform rotations around axes parallel to the x_{b}axis; - when point P lies on the x
_{b}z_{b}coordinate plane (i.e., y = 0), the three unit vectors**h**_{i}, for i = 1, 2, 3, (see Formulas (16)) are all parallel to the x_{b}z_{b}coordinate plane; therefore, the component of**ω**along**e**_{2}is not locked (see Equations (2)) and the platform can perform rotations around axes parallel to the y_{b}axis; - when point P lies on the x
_{b}y_{b}coordinate plane (i.e., z = 0), the three unit vectors**h**_{i}, for i = 1, 2, 3, (see Formulas (16)) are all parallel to the x_{b}y_{b}coordinate plane; therefore, the component of**ω**along**e**_{3}is not locked (see Equations (2)) and the platform can perform rotations around axes parallel to the z_{b}axis.

_{b}y

_{b}z

_{b}). Such a locus leaves eight wide simply-connected convex regions (i.e., the eight octants of Ox

_{b}y

_{b}z

_{b}) of the operational space, where the platform is constrained to translate. Inside any of these regions, the useful workspace of the studied 3-URU can be safely located. Moreover, since

**ω**= 0 in them, the instantaneous input–output relationship (i.e., system (5)) simplifies itself as follows

#### 3.1.2. Translation (Type-II(b)) Singularities of LaMaViP 3-URU

**V**, is different from zero. Therefore, the translation singularities are the configurations that satisfy the geometric condition

**V**) =

**v**

_{1}⋅(

**v**

_{2}×

**v**

_{3}) = 0

**v**

_{i}, for i = 1, 2, 3, are coplanar. This geometric condition occurs when the three segments B

_{i}C

_{i}, i = 1, 2, 3, (see Figure 1) are all parallel to a unique plane (see Figure 6). From an analytic point of view, the adopted notations (see Section 2 and Figure 1) bring to light the following relationships

**b**

_{i}−

**c**

_{i}) = r

_{i}

**v**

_{i}=

**p**+ (d

_{p}− d

_{b})

**e**

_{i}− f

_{i}(cosθ

_{i2}

**e**

_{i}+ sinθ

_{i2}

**h**

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

**p**(i.e.,

**p**= x

**e**

_{1}+ y

**e**

_{2}+ z

**e**

_{3}) and of

**h**

_{i}(i.e., Equations (16)), become

**b**

_{1}−

**c**

_{1}= [x + (d

_{p}− d

_{b}) − f

_{1}cosθ

_{12}]

**e**

_{1}+ [1 − f

_{1}m

_{1}sinθ

_{12}] y

**e**

_{2}+ [1 − f

_{1}m

_{1}sinθ

_{12}] z

**e**

_{3}

**b**

_{2}−

**c**

_{2}= [1 − f

_{2}m

_{2}sinθ

_{22}] x

**e**

_{1}+ [y + (d

_{p}− d

_{b}) − f

_{2}cosθ

_{22}]

**e**

_{2}+ [1 − f

_{2}m

_{2}sinθ

_{22}] z

**e**

_{3}

**b**

_{3}−

**c**

_{3}= [1 − f

_{3}m

_{3}sinθ

_{32}] x

**e**

_{1}+ [1 − f

_{3}m

_{3}sinθ

_{32}] y

**e**

_{2}+ [z + (d

_{p}− d

_{b}) − f

_{3}cosθ

_{32}]

**e**

_{3}

_{1}r

_{2}r

_{3}yields the equivalent equation

**b**

_{1}−

**c**

_{1}) ⋅ [(

**b**

_{2}−

**c**

_{2}) × (

**b**

_{3}−

**c**

_{3})] = 0

_{2}n

_{3}− n

_{1}n

_{2}− n

_{1}n

_{3}+ 2 n

_{1}n

_{2}n

_{3}]+ xy q

_{3}(1− n

_{1}n

_{2}) + xz q

_{2}(1 − n

_{1}n

_{3}) + yz q

_{1}(1 − n

_{2}n

_{3}) +

x q

_{2}q

_{3}+ y q

_{1}q

_{3}+ z q

_{1}q

_{2}+ q

_{1}q

_{2}q

_{3}= 0

_{1}= [1 − f

_{1}m

_{1}sinθ

_{12}]; n

_{2}= [1 − f

_{2}m

_{2}sinθ

_{22}]; n

_{3}= [1 − f

_{3}m

_{3}sinθ

_{32}]

_{1}= (d

_{p}− d

_{b}) − f

_{1}cosθ

_{12}; q

_{2}= (d

_{p}− d

_{b}) − f

_{2}cosθ

_{22}; q

_{3}= (d

_{p}− d

_{b}) − f

_{3}cosθ

_{32}

_{12}, θ

_{22}, and θ

_{32}, can be eliminated from Equation (25) by using the solution formulas of the inverse position analysis [18] reported in Appendix A. In doing so, Equation (25) becomes an equation that contains only the geometric constants of the machine and the platform pose coordinates, x, y, and z. Such equation, which is the analytic expression of a surface (the translation-singularity surface) in Ox

_{b}y

_{b}z

_{b}, can be exploited, during design, to determine the optimal values of the geometric constants of the machine that move the translation singularities into regions of the operational space which are far from the useful workspace.

#### 3.1.3. Serial (Type-I) Singularities of LaMaViP 3-URU

_{i2}, i = 1, 2, 3, since matrix

**G**is diagonal, and that the solution is indeterminate when at least one of the following geometric condition is satisfied (see Figure 1):

**g**

_{i}⋅ [(

**b**

_{i}−

**a**

_{i}) × (

**b**

_{i}−

**c**

_{i})] =

**g**

_{i}⋅ [(

**c**

_{i}−

**a**

_{i}) × (

**b**

_{i}−

**c**

_{i})] = f

_{i}r

_{i}sinθ

_{i3}= 0 i = 1,2,3

_{i3}= 0) or folded (θ

_{i3}= π). These two geometric conditions identify two concentric spherical surfaces with point A

_{i}as center, which point B

_{i}must lie on. From an analytic point of view, since

**b**

_{i}=

**p**+ d

_{p}

**e**

_{i}and

**a**

_{i}= d

_{b}

**e**

_{i}, the equations of these two spherical surfaces in Ox

_{b}y

_{b}z

_{b}can be written as follows (here, the square of a vector denotes the dot product of the vector by itself)

**b**

_{i}−

**a**

_{i})

^{2}= [

**p**+ (d

_{p}− d

_{b})

**e**

_{i}]

^{2}=

**p**

^{2}+ (d

_{p}− d

_{b})

^{2}+ 2 (d

_{p}− d

_{b})

**p**⋅

**e**

_{i}= (f

_{i}+ r

_{i})

^{2}i = 1,2,3

**b**

_{i}−

**a**

_{i})

^{2}= [

**p**+ (d

_{p}− d

_{b})

**e**

_{i}]

^{2}=

**p**

^{2}+ (d

_{p}− d

_{b})

^{2}+ 2 (d

_{p}− d

_{b})

**p**⋅

**e**

_{i}= (f

_{i}− r

_{i})

^{2}i = 1,2,3

_{1}− r

_{1})

^{2}≤ x

^{2}+ y

^{2}+ z

^{2}+ (d

_{p}− d

_{b})

^{2}+ 2 (d

_{p}− d

_{b}) x ≤ (f

_{1}+ r

_{1})

^{2}

_{2}− r

_{2})

^{2}≤ x

^{2}+ y

^{2}+ z

^{2}+ (d

_{p}− d

_{b})

^{2}+ 2 (d

_{p}− d

_{b}) y ≤ (f

_{2}+ r

_{2})

^{2}

_{3}− r

_{3})

^{2}≤ x

^{2}+ y

^{2}+ z

^{2}+ (d

_{p}− d

_{b})

^{2}+ 2 (d

_{p}− d

_{b}) z ≤ (f

_{3}+ r

_{3})

^{2}

_{b}= d

_{p}and f

_{i}= r

_{i}= R for i = 1, 2, 3, inequalities (29) give a sphere with center O and radius 2R as reachable workspace (see Figure 7).

#### 3.2. Singularity Analysis of the Actuation Device

**ω**=

**0**and of $\dot{\mathbf{p}}={\mathbf{V}}^{-1}\mathbf{G}\hspace{0.17em}{\dot{\mathsf{\theta}}}_{2}$ (see Equation (18)) into Equation (12) yields

**S**=

**I**

_{3 × 3}+

**KNMV**

^{−1}

**G**where,

**I**

_{3 × 3}is the 3 × 3 identity matrix, and

**S**= [s

_{ij}] gives the following explicit expression of its ij-th entry, s

_{ij}for i,j = 1, 2, 3,

_{ij}denotes the Kronecker delta and the subscript “(n+m) mod 3” denotes the sum with modulus 3 of the two integers n and m as defined in modular arithmetic [19].

**S**immediately reveals that, when matrix

**V**is not invertible (i.e., when Equation (20) is satisfied), relationship (32) is indeterminate. Such a condition does not provide further reductions of the regions where the useful workspace can be located since it coincides with the translation-singularity locus (i.e., with Equation (20)) analyzed in Section 3.1.2. Over this condition, Equation (32) fails to give unique values of the actuated-joint rates, ${\dot{\mathsf{\theta}}}_{\mathrm{i}2}$, i = 1, 2, 3, for assigned values of the angular velocities of the motor shafts, ${\dot{\mathsf{\theta}}}_{\mathrm{iM}}$, i = 1, 2, 3, when the determinant of matrix

**S**is equal to zero, that is, when the following geometric condition is satisfied

**S**) =

**s**

_{1}·(

**s**

_{2}×

**s**

_{3}) = 0

**s**

_{i}, for i = 1, 2, 3, are the column vectors of matrix

**S**. Therefore, an actuation singularity occurs when the three vectors

**s**

_{i}, for i = 1, 2, 3, are coplanar. From an analytic point of view, Equation (35) is the equation of a surface in Ox

_{b}y

_{b}z

_{b}, which corresponds to the actuation-singularity locus. Such equation can be put in the form f(x, y, z) = 0 by exploiting the above-reported expressions of the terms appearing in Equation (34) and can be used to size the geometric constants and the speed ratios k

_{i}, i = 1, 2, 3, so that the actuation singularity locus is far from the useful workspace.

**v**

_{i}and

**g**

_{i}. Differently from Equation (20), which is satisfied by the coplanarity of the three unit vectors

**v**

_{i}, i = 1, 2, 3, Equation (36) is satisfied by the coplanarity of these other three vectors that are not aligned with the unit vectors

**v**

_{i}, i = 1, 2, 3, any longer. Equation (36) can be put in the form f(x, y, z) = 0 by exploiting the above-reported expressions of the terms appearing in it and can be used as an alternative to Equations (20) and (35) to size the geometric constants and the speed ratios k

_{i}, i = 1, 2, 3, so that both the translation and the actuation singularity loci are far from the useful workspace.

## 4. Conclusions

## 5. Patents

## Funding

## Conflicts of Interest

## Appendix A. Inverse Position Analysis

_{12}, θ

_{22}, and θ

_{32}) for assigned values of the platform pose parameters (i.e., point P’s coordinates x, y, and z). This problem has been solved in [18]. In this appendix the solution illustrated in [18] is briefly summarized.

_{i}

^{2}+ β

_{i}

^{2}+ f

_{i}

^{2}− r

_{i}

^{2}− 2 f

_{i}(α

_{i}cos θ

_{i2}+ β

_{i}sin θ

_{i2}) = 0 i = 1,2,3

_{1}= x + d

_{p}− d

_{b}, α

_{2}= y + d

_{p}− d

_{b}, α

_{3}= z + d

_{p}− d

_{b}, β

_{1}= $\sqrt{{\mathrm{y}}^{2}+{\mathrm{z}}^{2}}$, β

_{2}= $\sqrt{{\mathrm{x}}^{2}+{\mathrm{z}}^{2}}$, β

_{3}= $\sqrt{{\mathrm{x}}^{2}+{\mathrm{y}}^{2}}$.

_{i2}= (1 − t

_{i}

^{2})/(1 + t

_{i}

^{2}) and sinθ

_{i2}= 2t

_{i}/(1 + t

_{i}

^{2}), where t

_{i}= tan(θ

_{i2}/2), into Equations (A1) transforms them into quadratic equations whose solutions are

_{i2}. From a geometric point of view, these two solutions per limb correspond to the up to two intersections of two circumferences that lie on the plane perpendicular to the unit vector

**g**

_{i}and passing through A

_{i}and B

_{i}, one with center at A

_{i}and radius f

_{i}and the other with center at B

_{i}and radius r

_{i}. These intersections are the possible positions of point C

_{i}(see Figure 1) compatible with an assigned platform pose.

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1 | Hereafter, P, R, S, and U stand for prismatic pair, revolute pair, spherical pair and universal joint. Additionally, the serial kinematic chains constituting the PM limbs are indicated by a string of such capital letters that give the sequence of joint types encountered by moving from the base to the platform along the considered limb. |

2 | It is worth noting that the parallelism of the coordinate axes is kept during the motion since the analyzed 3-URU is translational. |

3 | In Equation (3), the first equality is obtained by rearranging the kinematic relationship $\dot{\mathbf{p}}={\dot{\mathbf{b}}}_{\mathrm{i}}+\mathsf{\omega}\times (\mathbf{p}-{\mathbf{b}}_{\mathrm{i}})$ whereas, the last equality is deduced by introducing the kinematic relationship $\dot{\mathbf{c}}=({\dot{\mathsf{\theta}}}_{\mathrm{i}1}{\mathbf{e}}_{\mathrm{i}}+{\dot{\mathsf{\theta}}}_{\mathrm{i}2}{\mathbf{g}}_{\mathrm{i}})\times ({\mathbf{c}}_{\mathrm{i}}-{\mathbf{a}}_{\mathrm{i}})$ into the expression of the velocity of Bi when considered a point of the link CiBi (see Figure 1b), that is, $\dot{\mathbf{b}}={\dot{\mathbf{c}}}_{\mathrm{i}}+[{\dot{\mathsf{\theta}}}_{\mathrm{i}1}{\mathbf{e}}_{\mathrm{i}}+({\dot{\mathsf{\theta}}}_{\mathrm{i}2}+{\dot{\mathsf{\theta}}}_{\mathrm{i}3}\left){\mathbf{g}}_{\mathrm{i}}\right]\times ({\mathbf{b}}_{\mathrm{i}}-{\mathbf{c}}_{\mathrm{i}})$. |

4 | It is worth stressing that platform’s instantaneous DOFs may be different from the mechanism instantaneous DOFs since they depend on how effective are the mechanism constraints on the platform instantaneous motion and that they cannot exceed the DOF number of a free rigid body. |

5 | According to [17], here, the term “limb connectivity” denotes the DOF number the platform would have if it were connected to the base only through that limb. |

6 | It is worth reminding that the determinant of a 3 × 3 matrix is the mixed product of its three rows (or column) vectors. |

**Figure 1.**LaMaViP 3-URU with the R-pair axes that are fixed in the base (platform) mutually orthogonal: (

**a**) overall scheme and notations, (

**b**) detailed scheme of the i-th limb.

**Figure 2.**A possible mechanical transmission, based on a bevel gearbox, for actuating the 2nd R-pair of the i-th limb by keeping the actuator fixed to the base.

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**MDPI and ACS Style**

Di Gregorio, R.
A Novel 3-URU Architecture with Actuators on the Base: Kinematics and Singularity Analysis. *Robotics* **2020**, *9*, 60.
https://doi.org/10.3390/robotics9030060

**AMA Style**

Di Gregorio R.
A Novel 3-URU Architecture with Actuators on the Base: Kinematics and Singularity Analysis. *Robotics*. 2020; 9(3):60.
https://doi.org/10.3390/robotics9030060

**Chicago/Turabian Style**

Di Gregorio, Raffaele.
2020. "A Novel 3-URU Architecture with Actuators on the Base: Kinematics and Singularity Analysis" *Robotics* 9, no. 3: 60.
https://doi.org/10.3390/robotics9030060