In this section, the instantaneous input–output relationship of the LaMaViP 3-URU is deduced, and then, it is used for determining its singularity loci. The instantaneous input–output relationship is a linear mapping that relates the actuated-joint rates (instantaneous inputs), which, in the studied 3-URU, are ${\stackrel{.}{\mathsf{\theta}}}_{\mathrm{i}2}$, i = 1, 2, 3, and the platform twist (instantaneous outputs), that is, $\widehat{\mathbf{\$}}={\left({\dot{\mathbf{p}}}^{T},{\mathsf{\omega}}^{T}\right)}^{T}$ where **ω** is the angular velocity of the platform.

Equations (2) and (4) provide the following instantaneous input–output relationship for the LaMaViP 3-URU

where

**0**_{3 × 3} is the 3 × 3 null matrix,

and

Since the actuators are not directly mounted on the actuated joints, Equation (5) has to be accompanied by additional equations coming from the kinematic analysis of the actuation device (

Figure 2) in order to implement control algorithms. Such equations can be deduced as follows. With reference to

Figure 2 and

Figure 3, the following formulas can be stated

where

^{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

Eventually, let k

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

which yields

whose integration gives

where θ

_{iM|0} and θ

_{i1|0} are the values of θ

_{iM} and θ

_{i1}, respectively, when θ

_{i2} is equal to zero.

Equation (10) relates the actuated-joint rates to the angular velocities of the motor shafts and involves the non-actuated joint rates

${\dot{\mathsf{\theta}}}_{\mathrm{i}1}$, for i = 1, 2, 3. The dot product of Equation (3) by

**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:

System (13) is the direct relationship between the angular velocities of the motor shafts, ${\dot{\mathsf{\theta}}}_{\mathrm{iM}}$, for i = 1, 2, 3, and the platform twist, that is, it is the instantaneous-kinematics model necessary to the control system of the machine which replaces the first three equations of system (5).

#### 3.1. Singularity Analysis

The availability of the instantaneous input–output relationship allows the solution of two instantaneous-kinematics’ problems [

10]: the forward instantaneous-kinematics (FIK) problem and the inverse instantaneous-kinematics (IIK) problem. The FIK problem is the determination of the platform twist for assigned values of the actuated-joint rates; vice versa, the IIK problem is the determination of the actuated joint rates for an assigned value of the platform twist.

Singular configurations (singularities) are the PM configurations where one or the other or both of the two above-mentioned problems are indeterminate [

9,

10]. In particular [

9], type-I singularities refer to the indetermination of the IIK problem, type-II singularities refer to the indetermination of the FIK problem, and type-III singularities refer to the indetermination of both the two problems. From a kinematic point of view, type-I singularities correspond to limitations of the instantaneous mobility of the platform and are located at the workspace boundary; they are present in all the manipulators and are sometimes called “serial singularities”. Differently, type-II singularities are mainly inside the workspace and correspond either (a) to a local increase of platform’s instantaneous DOFs

4 or (b), without any local variation of platform’s instantaneous DOFs, to some platform DOFs that locally become non-controllable through the actuated joints (i.e., the physical constraints locally become no longer independent). They are present only in closed kinematic chains (i.e., in PMs) and are sometimes called “parallel singularities”.

Type-II(b) singularities may occur in any PM; whereas, type-II(a) singularities may occur only in lower-mobility PMs, whose limb connectivity

5 is higher than the PM DOFs. Type-II(a) singularities are named “constraint singularities” [

11] since the additional platform DOFs acquired at such singularities may make the platform change its type of motion (operating mode). In particular, in a TPM, such additional DOFs can only be instantaneous rotations which may make the platform exit from the pure-translation operating mode; that is why TPMs’ constraint singularities are also named “rotation singularities” and TPMs’ type-II(b) singularities are also named “translation singularities” [

8].

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

The platform translation is guaranteed if and only if the constraints applied to the platform by the three URU limbs make the platform angular velocity,

**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 condition

6Equation (14) is satisfied when the unit vectors

**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).

From an analytic point of view, the notations introduced in

Section 2 make it possible to write

and

Then, the introduction of the analytic expression of

**p** (i.e.,

**p** = x

**e**_{1} + y

**e**_{2} + z

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

Eventually, the introduction of the explicit expressions given by Equation (16) into the singularity condition (14) provides the following analytic equation of the geometric locus of the rotation (constraint) singularities

The analysis of Equation (17) reveals that the rotation singularity locus is constituted by the 3 coordinate planes x = 0, y = 0, and z = 0 (

Figure 5). Additionally, the analysis of

Figure 1, of Formula (16) and Equation (2) reveals that

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.

As a consequence, when P lies on a coordinate axis the platform locally acquires 2 rotational DOFs; whereas, when P coincides with O (i.e., x = y = z = 0) the platform locally acquires 3 rotational DOFs, even though the expression at the left-hand side of Equation (17) becomes indeterminate in all these cases.

In short, the rotation-singularity locus is constituted by three mutually orthogonal planes (i.e., the three coordinate planes of Ox

_{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

where

${\dot{\mathsf{\theta}}}_{2}={({\dot{\mathsf{\theta}}}_{12},{\dot{\mathsf{\theta}}}_{22},{\dot{\mathsf{\theta}}}_{32})}^{T}$; whereas, the instantaneous-kinematics model necessary to the machine control (i.e., system (14)) simplifies itself as follows

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

Out of constraint singularities, system (18) is the instantaneous input–output relationship to consider. With reference to system (18), the FIK is the determination of

$\dot{\mathbf{p}}$ for an assigned

${\dot{\mathsf{\theta}}}_{2}$. This problem has a unique solution if and only if the determinant of the coefficient matrix,

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

Equation (20) is satisfied when the unit vectors

**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

which, after the introduction of the analytic expressions of

**p** (i.e.,

**p** = x

**e**_{1} + y

**e**_{2} + z

**e**_{3}) and of

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

with

Eventually, the product of Equation (20) by the non-null constant r

_{1} r

_{2} r

_{3} yields the equivalent equation

which, after the introduction of Formulas (22a), (22b) and (22c), becomes the following analytic expression of the translation-singularity locus

where

The actuated-joint variables, θ

_{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

The solution of the IIK problem involves only the first three equations of system (5). The analysis of these three equations reveals that they can be separately solved with respect to θ

_{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):

The i-th Equation (27) is satisfied when the i-th limb is fully extended (θ

_{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)

Equation (28) are also the equations of the reachable-workspace boundaries. Therefore, the reachable workspace of the LaMaViP 3-URU can be analytically defined by the following system of inequalities

In the case d

_{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

Since the actuators are not directly mounted on the actuated joint in the LaMaViP 3-URU, the motion transmission must be analyzed to check whether there are configurations (hereafter called “actuation singularities”) in which the relationship (i.e., Equations (10)) between the actuated-joint rates, ${\dot{\mathsf{\theta}}}_{\mathrm{i}2}$, i = 1, 2, 3, and the angular velocities of the motor shafts, ${\dot{\mathsf{\theta}}}_{\mathrm{iM}}$, i = 1, 2, 3, is indeterminate. In this subsection, such relationship is deduced and analyzed.

The introduction of

**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

with

Then, the introduction of Equation (30) into Equation (10), after some rearrangements, gives the sought-after relationship between the actuated-joint rates, and the angular velocities of the motor shafts, that is,

with

${\dot{\mathsf{\theta}}}_{\mathrm{M}}=({{\dot{\mathsf{\theta}}}_{1\mathrm{M}},{\dot{\mathsf{\theta}}}_{2\mathrm{M}},{\dot{\mathsf{\theta}}}_{3\mathrm{M}})}^{T}$,

**S** =

**I**_{3 × 3} +

**KNMV**^{−1}**G** where,

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

The expansion of the above expression of matrix

**S** = [s

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

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

where δ

_{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].

The analysis of matrix

**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

where

**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.

From the point of view of the platform control, the presence of the actuation singularities justifies the difference between System (18) and System (19). In particular, unlike System (18), System (19) yields the following geometric expression of the translation-singularity locus

which imposes the zeroing of the mixed product of the three vectors that dot multiply

$\dot{\mathbf{p}}$ in the three equations of System (19). The i-th vector, for i = 1, 2, 3, of this vector triplet is associated to the i-th limb and lies on a plane spanned by the two unit vectors

**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.