Four-Dimensional Persistent Screw Systems of the General Type

Abstract

When a mechanism moves, the twist system $S$ of the end-effector generally varies. In significant special cases, $S$ is a subalgebra of the Lie algebra of the special Euclidean group, and it remains constant. In more general cases, $S$ remains invariant up to a proper isometry, thus preserving its class. A mechanism of this kind is said to generate a persistent screw system (PSS) of the end-effector. PSSs play an important role in mobility analysis and mechanism design. This paper presents the serial generators of $4$-dimensional PSSs with a constant class of the general type.

1 Introduction

Screw systems are the subspaces of the Lie algebra $se(3)$ of the Euclidean group $SE(3)$. Two screw systems are equivalent if one may be moved onto the other by a rigid-body displacement [4]. This equivalence relation divides the space of screw systems into infinitely many classes. The latter may be grouped into a finite number of general or special types [7, Chap. 12], within which the constituent classes are identified by the values of a small number of parameters. Screw systems belonging to the same class have the same dimension, type and shape, thus differing only in their pose in space.

For a mechanism in a configuration $\varvec{\varTheta }$, the possible instantaneous motions of the end-effector are given by a screw system $S(\varvec{\varTheta })\subset {se(3)}$. In general, when $\varvec{\varTheta }$ changes, so does $S(\varvec{\varTheta })$. An important special case occurs when the mechanism constrains the body to trace out (at least locally) a subgroup of $SE(3)$. Then, for any nonsingular $\varvec{\varTheta }$, $S(\varvec{\varTheta })=A$, where $A$ is the algebra of the subgroup. The mechanism is said to generate an invariant screw system (ISS) of the end-effector [4–7]. Herein, a more general case is considered, namely a mechanism where $S(\varvec{\varTheta })$, although not necessarily constant, has a constant class . In other terms, $S(\varvec{\varTheta })$ retains its shape while it changes its pose, in effect moving like a rigid body in space. If that is so, and $\dim {S(\varvec{\varTheta })}=n$, the mechanism is said to generate an $n$-dimensional persistent screw system of the end-effector, or briefly an $n$-PSS. PSSs were presented by Carricato and Rico Martínez, who showed that PSS generators may be obtained by serially composing generators of ISSs [1–3]. The exhaustive classification of PSS generators is in progress. This paper complements the results in [1], where the generators of $4$-PSSs with constant classes of special types were described. Here, the serial generators of $4$-PSSs with constant classes of the general type are presented.

In the following, the locutions ‘$nG$ system’ and ‘$nR$ system’, with $R$ being a Roman numeral, denote $n$-dimensional screw systems, respectively, of the general type and of the $R$th special type, according to [7]. A normalized screw representing a relative twist between two bodies is designated by $\mathbf{S}$. The axis (when it exists), the pitch and the direction of $\mathbf{S}$ are denoted by $\ell$, $h$ and $\mathbf s$, respectively. When it is useful, $\ell$ and $h$ accompany $\mathbf{S}$ within parentheses, i.e. $\mathbf{S}(\ell ,h)$ (if $h=\infty$, $\ell$ is replaced by $\mathbf{s}$). Given two screws $\mathbf{S}_i(\ell _i,h_i)$ and $\mathbf{S}_j(\ell _j,h_j)$ (Fig. 1a), $n_{ij}$ is the common normal between $\ell _i$ and $\ell _j$; $P_{ij,i}$ and $P_{ij,j}$ are the feet of $n_{ij}$ on $\ell _i$ and $\ell _j$; $\mathbf{n}_{ij}$ is a unit vector parallel to $n_{ij}$ and directed from $P_{ij,i}$ to $P_{ij,j}$; $p_{ij}$ and $\alpha _{ij}$ are the shortest distance and the relative angle between $\ell _i$ and $\ell _j$, with $\alpha _{ij}$ being measured about $\mathbf{n}_{ij}$ in the interval $(-\pi /2,\pi /2]$; finally, $\mathbf{p}_{ij}=P_{ij,j}-P_{ij,i}=p_{ij}\mathbf{n}_{ij}$.

Fig. 1

Relative pose between two screws (a); a chain of four $1$-dof lower pairs (b)

2 Generators of 4-Dimensional PSSs of the General Type

Any subgroup of $SE(3)$ may be generated (at least locally) by a serial chain composed by $1$-dof lower pairs. Hence, any $4$-PSS generator emerging by the serial composition of ISSs may be considered ‘equivalent’ to a serial linkage $\mathcal S$ composed by (at least) four $1$-dof joints (Fig. 1b). Since joint motions affect neither the pitches nor the relative pose of adjacent joints, the geometry of the chain is completely defined by the joint screws at an arbitrarily-chosen nonsingular reference configuration, i.e. $\mathbf{S}_i = \left. \mathbf{S}_i(\varvec{\varTheta })\right| _{\varvec{\varTheta }=\mathbf{0}} = \mathbf{S}_i(\mathbf{0})$, $i=1\ldots 4$. Accordingly, $\mathcal S$ may be identified with the array $\langle \mathbf{S}_1,\ldots ,\mathbf{S}_4\rangle$. Link $0$ of $\mathcal S$ is the predecessor of $\mathbf{S}_1$; link $i$, with $i=1\ldots 3$, is the body laid between $\mathbf{S}_i$ and $\mathbf{S}_{i+1}$; and link $4$ is the successor of $\mathbf{S}_4$.

After a displacement $\varvec{\varTheta }=(\theta _1,\ldots ,\theta _4)$, the $i$th joint screw, $i > 1$, is moved to $\mathbf{S}_i(\varvec{\varTheta }) = \prod _{j=1}^{i-1} {\mathbf{D}_j}\left( \theta _j\right) {\mathbf{S}_i}$, where $\mathbf{D}_{j}$ is the adjoint action of the $j$th joint displacement. Generally, in the new configuration $S(\varvec{\varTheta })={{\mathrm{span}}}\left[ \mathbf{S}_1(\varvec{\varTheta }),\ldots ,\mathbf{S}_4(\varvec{\varTheta })\right] \ne S(\mathbf{0})$. $\mathcal S$ generates a PSS if, for every nonsingular $\varvec{\varTheta }$, $S(\varvec{\varTheta })=\mathbf{G}(\varvec{\varTheta })S(\mathbf{0})$, where $\mathbf{G}(\varvec{\varTheta })$ is the adjoint action of a proper isometry. A simpler formulation of this condition emerges by observing that joint motions $\theta _1$ and $\theta _4$ cannot alter the shape of $S(\varvec{\varTheta })$, since they do not affect the relative pose of $\mathcal S$’s joint screws. Accordingly, when studying the persistent properties of $S(\varvec{\varTheta })$, $\theta _1$ and $\theta _4$ may be kept constant, e.g. $\theta _1=\theta _4=0$, and link $2$ may be conveniently chosen as the reference frame [2]. As a consequence, $\mathcal S$ generates a $4$-PSS if and only if, for every nonsingular pair $(\theta _2,\theta _3)$, there is an adjoint action $\mathbf{G}(\theta _2,\theta _3)$ such that $S(\theta _2,\theta _3) = \mathbf{G}(\theta _2,\theta _3){S(0,0)}$, where $S(\theta _2,\theta _3)={{\mathrm{span}}}\left[ \mathbf{D}_2^{-1}(\theta _2)\mathbf{S}_1,\mathbf{S}_2,\mathbf{S}_3,\mathbf{D}_3(\theta _3)\mathbf{S}_4\right]$.

$S(\varvec{\varTheta })$ has a constant class of the general type if the principal screws $\mathbf{S}_{r1}(\ell _{p1},-h_{p1})$ and $\mathbf{S}_{r2}(\ell _{p2},-h_{p2})$ of the cylindroid $S^{\perp }$ reciprocal to $S$ have constant finite pitches, and $h_{p1}\ne {h_{p2}}$ [7] (Fig. 2). If that is so, the principal screw $\mathbf{S}_{pi}(\ell _{pi},h_{pi})$ of $S$, with $i=1,2$, is collinear with $\mathbf{S}_{ri}$ and has pitch $h_{pi}$, whereas the principal screws $\mathbf{S}_{p3}$ and $\mathbf{S}_{p4}$ span a cylindrical ISS $\mathcal{C }(\ell _{p3})$ along the nodal line of $S^{\perp }$ (the latter is the line perpendicular to $\ell _{p1}$ and $\ell _{p2}$, passing through their intersection point $O$). $\mathcal{C }(\ell _{p3})$ and the $\infty$-pitch screw therein are, respectively, the only available ISS with dimension greater than $1$ and the only $\infty$-pitch screw in $S$.

A generator $\mathcal S =\langle \mathbf{S}_1,\ldots ,\mathbf{S}_4\rangle$ of a $4G$-PSS may be constructed by composing the ISSs available in $S$. Since $\mathcal{C }(\ell _{p3})$ has dimension $2$ and $S$ has dimension $4$, no less than three ISSs need to be composed. A ternary generator is constructed by composing $\mathcal{C }(\ell _{p3})$ with two $1$-dimensional ISSs, i.e. with two single screws. A quaternary generator emerges by composing four distinct screws in $S$. Since the $4G$ system comprises a single $\infty$-pitch screw, $\mathcal S$ cannot include: more than one $\infty$-pitch screw; more than a pair of adjacent parallel finite-pitch screws; a pair of adjacent parallel finite-pitch screws plus an $\infty$-pitch screw. Otherwise, configurations in which $\infty$-pitch screws along more than one direction would appear.

$\mathcal S$ may be synthesized by expanding the $3$-PSS generators disclosed in [3]. Since the only $3$-systems contained in a $4G$ system are the $3G$, $3I$, $3III$, $3VII$ and $3VIII$ systems, the only $3$-PSSs that may appear within a $4G$ system are the $3I$- and the $3VIII$-PSS (cf. [3]). For this reason, these two will be the ‘building blocks’ of the generators described hereafter.

Fig. 2

Principal screws of a $4$-system of the general type: $h_{p1}\ne \infty \ne {h_{p2}}$, and $h_{p1}\ne h_{p2}$

Fig. 3

$3I$ systems within a $4G$ system

2.1 The Ternary Generator

Let $\mathcal B =\left\langle \mathbf{S}_2(\ell _2,h_2),\mathbf{S}_3(\ell _3,h_3),\mathbf{S}_4(\ell _4,h_4)\right\rangle$ and let $\mathcal B$ form a $3I$-PSS $B$. For the properties of the $3I$ systems, all screws of $B$ have finite pitch and, if $\mathbf{S}_{pi}^B(\ell _{pi}^B,h_{pi}^B)$, $i=1\ldots 3$, is the $i$th principal screw of $B$, then $h_{p1}^B=h_{p2}^B$ [7]. Since $B$ is persistent, the following conditions also apply [3]:

$$\begin{aligned} h_3=h_{p3}^B=0,\quad \mathbf{S}_3(\ell _3,0)=\mathbf{S}_{p3}^B(\ell _{p3}^B,0),\quad h_{p1}^B=h_{p2}^B\ne 0,\quad {P_{32,3}}\equiv {P_{34,3}}\equiv {O^B}, \end{aligned}$$

(1)

where $O^B$ is the point where $\mathbf{S}_{p1}^B$, $\mathbf{S}_{p2}^B$ and $\mathbf{S}_{p3}^B$ intersect. Furthermore, for $j=2,4$,

$$\begin{aligned} \alpha _{3j}\ne 0,\quad h_j = {h_{p1}^B}{\sin ^2}{\alpha _{3j}} \ne 0,\quad {p_{3j}} = {h_{p1}^B}\sin {\alpha _{3j}}\cos {\alpha _{3j}}. \end{aligned}$$

(2)

It may be proven that any $3I$ system lying within a $4G$ system must meet the following requirements (where $h_{p3}^B=0$ is enforced, Fig. 3):

• (1) $\mathbf{S}_{p1}^B=\mathbf{S}_{p1}$, i.e. $\ell _{p1}^B\equiv \ell _{p1}$ and $h_{p1}^B=h_{p2}^B=h_{p1}$, with $\mathbf{S}_{p1}$ being the principal screw normal to $\ell _{p3}$ with the highest pitch in absolute value, i.e. $|h_{p1}|\ge |h_{p2}|$;

• (2) the center $O^B$ of $B$ must lie on $\ell _{p1}$, namely $O^B-O = r^B \mathbf{s}_{p1}$, with $r^B\in \mathbb R$;

• (3) $\cos ^2{\alpha ^B} = (h_{p2}-h_{p3}^B)/(h_{p1}-h_{p3}^B) = h_{p2}/h_{p1}$, with $\alpha ^B$ being the angle that $\ell ^B_{p3}$ forms with $\ell _{p3}$, evaluated according to the right-hand rule about $\mathbf{s}_{p1}$ and such that $-\pi /2<\alpha ^B\le \pi /2$;

• (4) $r^B = (h_{p1}-h_{p3}^B) \sin \alpha ^B \cos \alpha ^B = h_{p1} \sin \alpha ^B \cos \alpha ^B$;

• (5) $h_{p1}$ and $h_{p2}$ must have the same sign (when they are different from zero), i.e. $h_{p1}\ge h_{p2}\ge h_{p3}^B = 0$ or $h_{p1}\le h_{p2}\le h_{p3}^B = 0$.

For the sake of brevity, the proof of the above statements is not reported. It emerges from statements (1)–(5) that a $4G$ system may comprise only two $3I$ systems, i.e. $B$ and $B'$, such that $h_{p3}^B=0$ (Fig. 3). $B$ and $B'$ are symmetric under reflection in $\ell _{p3}$, and they coalesce when $h_{p2}=0$ (in which case, $\alpha ^B=\pi /2$ and $r^B=0$).

Fig. 4

Ternary generator of a $4$-PSS with a constant class of the general type

If $S(\varvec{\varTheta })$ has to be a $4G$-PSS, $h_{p1}$ and $h_{p2}$ must remain constant as $\varvec{\varTheta }$ varies. Hence, $\alpha ^B$ must remain constant too. Two cases need to be distinguished, depending on whether $\mathbf{S}_1$ has infinite or finite pitch. If $h_1=\infty$, $\mathbf{S}_1$ must be parallel to ${\ell _{p3}}$, as the only $\infty$-pitch screw of $S$ lies in $\mathcal C (\ell _{p3})$. Since $\ell ^B_{p3}\equiv {\ell _3}$, $\alpha ^B$ must thus coincide with the angle $\alpha _{31}$ between $\mathbf{s}_1$ and $\mathbf{s}_3$. Since $\cos \alpha _{31} = \mathbf{s}_1\cdot \mathbf{s}_3 = \cos \alpha _{32}\cos \alpha _{21} - \sin \alpha _{32}\sin \alpha _{21}\cos \theta _2$ and $\sin \alpha _{32}\ne 0$, $\alpha _B$ may be constant only if $\alpha _{21}=0$, in which case $\alpha ^B=\alpha _{31}=\alpha _{32}$. Hence, $\mathbf{S}_1$ and $\mathbf{S}_2$ are parallel, and they span $\mathcal C (\ell _{p3})$ (Fig. 4). This condition must be enforced also if $h_1$ is finite. In fact, $\mathbf{S}_1$ must be reciprocal, for arbitrary values of $\theta _2$, to a screw $\mathbf{S}_{r1}$ of pitch $-h_{p1}$ passing through $O^B$ and lying on a plane perpendicular to $\ell _3$. It is not difficult to verify, by direct computation, that this may happen only if $\mathbf{S}_1$ and $\mathbf{S}_2$ are collinear, i.e. if they span $\mathcal C (\ell _{p3})$.¹ The $4$-system illustrated in Fig. 4 is, thus, persistent. $O$ coincides with $P_{32,2}$, $\mathbf{S}_{p1}$ lies along the common normal between $\mathbf{S}_2$ and $\mathbf{S}_3$, $\mathbf{S}_{p2}$ is orthogonal to $\ell _{p1}$ and $\ell _{p3}$, and the following conditions (deriving from Eqs. (1) and (2) and statements (1)–(3) apply:

$$\begin{aligned}&h_3 = 0,\quad h_4 = h_{p1}\sin ^2{\alpha _{34}},\quad p_{34} = h_{p1}\sin {\alpha _{34}}\cos {\alpha _{34}}, \end{aligned}$$

(3a)

$$\begin{aligned}&p_{32} = h_{p1}\cos {\alpha _{32}}\sin {\alpha _{32}},\quad h_{p2} = h_{p1}{\cos ^2}{\alpha _{23}}, \end{aligned}$$

(3b)

with $h_{p1}\ge h_{p2}\ge 0$ or $h_{p1}\le h_{p2}\le 0$. In particular, when $h_{p2}=0$ and $h_{p1}\ne 0$, Eq. (3b) implies $\alpha _{32}=\pi /2$ and $p_{32}=0$. When $h_{p1}=0$, Eq. (3a, b) require $h_{p2}=h_4=p_{32}=p_{34}=0$, and a ternary generator of a $4I$-PSS is indeed obtained (cf. Fig. 2c in [1]).

The $4G$-PSS in Fig. 4 may also be derived from the $3VIII$-PSS generator shown in Fig. 7a of [3]. Let $\mathcal A =\left\langle \mathbf{S}_1(\mathbf{s}_1,\infty ),\mathbf{S}_2(\ell _2,h_2),\mathbf{S}_3(\ell _3,h_3)\right\rangle$, where $\mathbf{S}_1(\mathbf{s}_1,\infty )$ and $\mathbf{S}_2(\ell _2,h_2)$ form a cylindrical ISS $\mathcal C (\ell _2)$, and $\mathbf{S}_3(\ell _3,h_3)$ is a finite-pitch screw nonparallel to $\ell _2$. According to [3], $\mathcal A$ generates a $3VIII$-PSS $A$ such that

$$\begin{aligned} h_{p2}^A = h_3 + p_{32}\cot {\alpha _{32}}, \end{aligned}$$

(4)

where $\mathbf{S}_{p2}^A(\ell _{p2}^A,h_{p2}^A)$ is the principal screw of $A$ perpendicular to $\ell _2$ and $\mathbf{n}_{32}$. Since $A$ contains one $\infty$-pitch screw and $\mathcal S =\left\langle \mathcal A ,\mathbf{S}_4(\ell _4,h_4)\right\rangle$ has to generate a $4G$-PSS, $\mathbf{S}_4$ must have finite pitch, and it cannot be parallel to $\ell _3$.

Fig. 5

Ternary generator of a $4G$-PSS obtained from a $3VIII$-PSS

If $\theta _3$ is the angle between $\mathbf{n}_{32}$ and $\mathbf{n}_{34}$, whenever $\theta _3=k\pi$, $k\in \mathbb N$, $\ell _2$, $\ell _3$ and $\ell _4$ are perpendicular to the same direction, i.e. $\mathbf{n}_{32}=\pm \mathbf{n}_{34}$ (Fig. 5). In such configurations $\mathcal C (\ell _2)$, $\mathbf{S}_3$ and $\mathbf{D}_3(k\pi )\mathbf{S}_4$ cannot be linearly independent, otherwise they would form a $4$-system reciprocal to an $\infty$-pitch screw parallel to $\mathbf{n}_{32}$ (and this could not be a $4G$ system). Thus, these configurations must be singular and, therein, $\mathbf{D}_3(k\pi )\mathbf{S}_4$ must belong, for any $k$, to $A$. This implies $P_{32,3}=P_{34,3}$ and $h_3=0$. Moreover, since $\mathbf{D}_3(0)\mathbf{S}_4$ and $\mathbf{D}_3(\pi )\mathbf{S}_4$ are symmetric under reflection in $\ell _3$, they must form a cylindroid $C$ having one of its principal screws, say $\mathbf{S}_{p2}^C$, on $\ell _3$ [7, Chap. 3]. Since $C$ belongs to $A$ and no line in $A$ other than $\ell _2$ may contain more than one screw, $\mathbf{S}_{p2}^C$ must necessarily coincide with $\mathbf{S}_3$, so that $h_{p2}^C=h_3=0$. By letting $\mathbf{S}_{p1}^C(\ell _{p1}^C,h_{p1}^C)$ be the other principal screw of $C$, it must be [7, Chap. 3]

$$\begin{aligned}&h_4 = h_{p2}^C + (h_{p1}^C-h_{p2}^C)\sin ^2{\alpha _{34}} = h_{p1}^C\sin ^2{\alpha _{34}}, \end{aligned}$$

(5a)

$$\begin{aligned}&p_{34} = (h_{p1}^C-h_{p2}^C)\sin {\alpha _{34}}\cos {\alpha _{34}} = h_{p1}^C\sin {\alpha _{34}}\cos {\alpha _{34}}. \end{aligned}$$

(5b)

By requiring $\mathbf{S}_{p1}^C$ to belong to $A$, one also obtains that

$$\begin{aligned} h_{p2}^A = h_{p1}^C + p_{32}\cot ({\alpha _{32}+\pi /2}) = h_{p1}^C - p_{32}\tan {\alpha _{32}} \end{aligned}$$

(6)

and, thus, recalling Eq. (4) and rearranging terms,

$$\begin{aligned} p_{32} = h_{p1}^C\sin {\alpha _{32}}\cos {\alpha _{32}},\quad h_{p2}^A = h_{p1}^C{\cos ^2}{\alpha _{32}}. \end{aligned}$$

(7)

By varying $\theta _3$, $\mathbf{S}_3$ and $\mathbf{D}_3(\theta _3)\mathbf{S}_4$ generate a pencil of cylindroids all congruent to $C$, i.e. a $3I$-PSS with principal pitches equal to $0$ and $h^C_{p1}$. By letting $h^C_{p1}=h_{p1}$ and $h^A_{p2}=h_{p2}$, Eqs. (5) and (7) coincide with Eq. (3a, b). The generator in Fig. 4 is evidently re-obtained.

Fig. 6

Quaternary generator of a $4$-PSS with a constant class of the general type

2.2 The Quaternary Generator

The arguments developed in Sect. 2.1 provide a clue for obtaining a quaternary generator of $4G$-PSS. It has been seen, in fact, that a $4G$ system such that $h_{p1}>h_{p2}\ge 0$ or $h_{p1}<h_{p2}\le 0$ comprises only two $3I$ systems with a $0$-pitch central principal screw, i.e. $B$ and $B'$ (Fig. 3). According to statements (2)–(4), the poses of the central axes $\ell ^B_{p3}$ and ${\ell ^B_{p3}}'$ of $B$ and $B'$ are unambiguously determined by the value of the principal pitches of $S$, i.e. $h_{p1}$ and $h_{p2}$. In Fig. 4, the screws $\mathbf{S}_3$ and $\mathbf{D}_3(\theta _3)\mathbf{S}_4$ span (as $\theta _3$ varies) one of these $3I$ systems, say $B$, with the entire $4$-system being generated by composing $B$ with the cylindrical ISS $\mathcal C (\ell _{p3})$. If $\mathcal C (\ell _{p3})$ is replaced by two screws, i.e. $\mathbf{S}_2(\ell _2,0)$ and $\mathbf{D}_2^{-1}(\theta _2)\mathbf{S}_1(\ell _1,h_1)$, which span (as $\theta _2$ varies) the other $3I$ system, i.e. $B'$, the same vector subspace is obviously obtained (Fig. 6). The joint screws of the described generator satisfy the conditions

$$\begin{aligned}&h_2=h_3=0,\quad \alpha _{p3,2}=\alpha _{p3,3}, \end{aligned}$$

(8a)

$$\begin{aligned}&h_{p2}=h_{p1}\cos ^2{\alpha _{p3,2}},\quad p_{p3,2}=p_{p3,3}=h_{p1}\cos {\alpha _{p3,2}}\sin {\alpha _{p3,2}}, \end{aligned}$$

(8b)

$$\begin{aligned}&h_1=h_{p1}\sin ^2{\alpha _{21}},\quad p_{21}=h_{p1}\sin {\alpha _{21}}\cos {\alpha _{21}}, \end{aligned}$$

(8c)

$$\begin{aligned}&h_4=h_{p1}\sin ^2{\alpha _{34}},\quad p_{34}=h_{p1}\sin {\alpha _{34}}\cos {\alpha _{34}}, \end{aligned}$$

(8d)

where $p_{p3,i}$ and $\alpha _{p3,i}$ are, respectively, the shortest distance and the relative angle between $\ell _i$ and $\ell _{p3}$, $i=2,3$. The quaternary generator in Fig. 6 does not allow $h_{p2}=0$ and $h_{p1}\ne 0$, since in this case $\alpha _{p3,2}=\alpha _{p3,3}=\pi /2$ and $p_{p3,2}=p_{p3,3}=0$, and $\mathbf{S}_2$ and $\mathbf{S}_3$ would coincide. Also, this generator may not degenerate into a $4I$-PSS, as for $h_{p1}=0$ all screws would have zero pitch and pass through $O$.

3 Conclusions

Two generators of $4$-dimensional screw systems with a constant class of the general type (i.e. $4G$-PSSs) were disclosed. It may be proven that no other $4G$-PSS generators exist. Due to space limitations, the proof is omitted, but it will be reported in a future extended version of the contribution.