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 , i = 1, 2, 3, and the platform twist (instantaneous outputs), that is, where ω is the angular velocity of the platform.
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 ωpq 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
Equation (10) relates the actuated-joint rates to the angular velocities of the motor shafts and involves the non-actuated joint rates
, for i = 1, 2, 3. The dot product of Equation (3) by
gi, after some algebraic manipulations, relates the joint rates
, 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, , 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
hi, for i = 1, 2, 3, are coplanar. Since the i-th unit vector
hi is perpendicular to the plane passing through the coordinate axis of Ox
by
bz
b with the direction of
ei where the unit vector
gi 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
e1 + y
e2 + z
e3) 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 ybzb coordinate plane (i.e., x = 0), the three unit vectors hi, for i = 1, 2, 3, (see Formulas (16)) are all parallel to the ybzb coordinate plane; therefore, the component of ω along e1 is not locked (see Equations (2)) and the platform can perform rotations around axes parallel to the xb axis;
when point P lies on the xbzb coordinate plane (i.e., y = 0), the three unit vectors hi, for i = 1, 2, 3, (see Formulas (16)) are all parallel to the xbzb coordinate plane; therefore, the component of ω along e2 is not locked (see Equations (2)) and the platform can perform rotations around axes parallel to the yb axis;
when point P lies on the xbyb coordinate plane (i.e., z = 0), the three unit vectors hi, for i = 1, 2, 3, (see Formulas (16)) are all parallel to the xbyb coordinate plane; therefore, the component of ω along e3 is not locked (see Equations (2)) and the platform can perform rotations around axes parallel to the zb 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
by
bz
b). Such a locus leaves eight wide simply-connected convex regions (i.e., the eight octants of Ox
by
bz
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
; 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
for an assigned
. 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
vi, for i = 1, 2, 3, are coplanar. This geometric condition occurs when the three segments B
iC
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
e1 + y
e2 + z
e3) and of
hi (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
by
bz
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
bi =
p + d
pei and
ai = d
bei, the equations of these two spherical surfaces in Ox
by
bz
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, , i = 1, 2, 3, and the angular velocities of the motor shafts, , i = 1, 2, 3, is indeterminate. In this subsection, such relationship is deduced and analyzed.
The introduction of
ω =
0 and of
(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
,
S =
I3 × 3 +
KNMV−1G where,
I3 × 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,
, i = 1, 2, 3, for assigned values of the angular velocities of the motor shafts,
, i = 1, 2, 3, when the determinant of matrix
S is equal to zero, that is, when the following geometric condition is satisfied
where
si, for i = 1, 2, 3, are the column vectors of matrix
S. Therefore, an actuation singularity occurs when the three vectors
si, for i = 1, 2, 3, are coplanar. From an analytic point of view, Equation (35) is the equation of a surface in Ox
by
bz
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
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
vi and
gi. Differently from Equation (20), which is satisfied by the coplanarity of the three unit vectors
vi, i = 1, 2, 3, Equation (36) is satisfied by the coplanarity of these other three vectors that are not aligned with the unit vectors
vi, 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.