1. Introduction
When planning the motion of lower-mobility parallel mechanisms, we may need to plan the distribution of a serial of instantaneous screw axes (ISAs) of the moving platform (i.e., to plan its axode) in the reference frame of the mechanisms for some special application scenarios. However, the rotational axes of lower-mobility parallel mechanisms cannot be chosen arbitrarily in the reference frame of the mechanisms [
1]. For the above fact accounts the corresponding lower-order screw system (the order of screw system is no more than three) [
2] whose screw axes distribute in a finite zone of Euclidean three-space. So, this finite zone must be taken into consideration when we plan the axode of lower-mobility parallel mechanisms.
Although Ball and Hunt [
3,
4,
5] have revealed that the equal-pitch screws of the general three-system lie on a pitch-hyperboloid, there have been few papers providing a deeper and more comprehensive study on the finite zone and the distribution characteristics of the
screw axes of the general three-system in Euclidean three-space. To provide some useful guides for the axode planning of three degree-of-freedom parallel mechanisms, we originally present the mathematical expressions of the finite distribution zone of the
screw axes of the general three-system by deriving the
striction points on a screw axis when it is infinitely close to its surrounding
screw axes in its infinitesimal neighborhood space. Furthermore, we originally reveal the distribution regularity of the
screw axes of the general three-system by partitioning them into
two-subsystems [
6].
For the convenience of analysis of the distribution characteristics of the general three-system, we compute and derive it in its principal frame [
4,
7] in which its three principal screws attain their simplest forms because they are mutually and orthogonally concurrent at the origin of the principal frame and because their screw axes are along the
x-,
y-, and
z-axes of the principal frame, respectively. When planning the axode of lower-mobility parallel mechanisms, we need to transform the distribution zone to the reference frame of mechanisms. So, identifying the principal screws of a screw system when a set of general basis screws has been provided is a key issue in axode planning. This problem has been reduced to an eigenproblem by Parkin [
8] (1990), Tsai [
9] (1993), Parkin [
10] (1997), Rico [
11] (1998), Bandyopadhyay [
12] (2004), Bandyopadhyay [
13] (2009), and Zhao [
14] (2009). In particular, Zhang [
15] (1998) proposed an algebraic method computing the principal screws of three-systems according to the relationship between two-systems and three-systems. Fang [
16] (1998) obtained the principal screws of three-systems by employing the conical section degenerating theory. Bandyopadhyay [
13] pointed out that the solution of the generalized eigenproblem gives rise to the extreme values of the pitch by differentiating the expression of the pitch and arriving at the identical equation with the eigenproblem.
On the distribution of screw axes of the screw system, Huang [
17] discussed the correspondence between the finite screw system and projective space. Inspired by the one-to-one mapping relationship [
3] between the points on a unit circle and the screws within a two-system, Zlatanov [
6] developed the one-to-one mapping between the points on a unit sphere and the screws within a three-system. Study’s map [
18] clarifies there is a one-to-one correspondence between oriented straight lines in Euclidean three-space and the dual points on the surface of a dual unit sphere in dual space. Hence, a differential curve on the sphere corresponds to a ruled surface in line space.
Since the above references study the distribution of screw axes in abstract spaces, it is hard to apply them to axode planning in Euclidean three-space. To develop intuitive and visual methods of axode planning, we especially study the distribution regularity of the screw axes of the general three-system in Euclidean three-space. Furthermore, to build a bridge between axode planning and the motion planning of parallel mechanisms, we have developed a kinematic model of screw rolling between fixed and moving axodes [
19,
20], which solves the problem, “Given a pair of fixed and moving axodes, how to reproduce the associated continuous rigid motion?” For the distribution zone of the screw axes of higher-order screw systems, perhaps it can be decomposed into that of lower-order screw systems, as Chen [
2] has decomposed the four-, five-, and six-systems into the direct sums of several two- and three-systems by the decomposition of the positive semidefinite (PSD) method.
The advances of screw theory in mechanisms are briefly listed here. Toscano [
21] developed the kinetostatic model for HRs based on the screw theory. Zhao [
22] proved that the time derivative of the twist is a screw and revealed that its physical meaning is the rigid-body acceleration. Sharafian [
23] enhanced the joint-based method to a new version by resorting to reciprocal screws, and introduced the constraint forces and moments distribution indices. Gallardo [
24] carried out the kinematic and dynamic analyses of the 2 (3-RPS) series-parallel manipulator by employing the theory of screws and the principle of virtual work. Fan [
25] proposed a novel concept of a virtual screw that is formulated by a combination of virtual angular velocity and virtual linear velocity. Bu [
26] developed a novel metric based on the geometric average normalized volume spanned by weighted screws to measure the distance from the current configuration to a singular configuration. Gallardo [
27] obtained the input–output equations of velocity and acceleration of the 5-RPUR parallel mechanism by resorting to reciprocal screws.
The remainder of this article is organized as follows.
Section 2 discusses the distribution of the position vector endpoints of the
screw axes of the general three-system. Since the position endpoints are the particular case of the striction points on the screw axis,
Section 3 studies the distribution of the
striction points of the
screw axes. The above two sections exactly describe the finite distribution zone of screw axes of the general three-system in Euclidean three-space. To reveal the inner structure of this finite zone,
Section 4 decomposes the general three-system into
two-subsystems and presents the detailed distribution regularity of
cylindroids fulfilling the finite zone by three theorems. In pursuit of a generalized decomposition method,
Section 5 proposes the varying-pitch ruled surface based on the pitch-hyperboloid. A numerical example of the 3-RPS pyramid mechanism is presented for the application of the concept ‘the densest distribution zone’ in
Section 6.
2. Distribution Surface of Position Vector Endpoints of Screw Axes
In this section, we mainly discuss the positions of the
screw axes of the general three-system. The general three-system of screws was firstly proposed by Hunt [
4] when he classified the screw systems by the pitch values of their principal screws. In the principal frame, three principal screws of general three-systems can be expressed, respectively, as:
in the form of unit Plücker coordinates, where the superscript
p represents principal screws. Without a loss of generality, we take the three principal pitches as
because they are mutually distinct and are finite real numbers. Consequently, any screw within this screw system has the form
where the coefficients
and
are both real numbers.
Remark: Thescrews represented by the equation are collinear, but have opposite directions with those represented by Equation(2), respectively. Since they are mirror images, they have the same distribution characteristics in Euclidean three-space. So, we only confine attention to Equation(2) in this paper.
Setting the primary part
of
and the dual part
of
, the pitch of
has the form
We make a segment
from the origin
O of the principal frame to the point
P on the axis of
. For the uniqueness of the position representation of the screw
,
P is the nearest point to the origin
O among the
points on the axis of
. In other words, the segment
is perpendicular to the axis of
. The vector
is defined as the position vector of the screw
with respect to the principal frame and has the form
The perpendicular foot
P is defined as the position vector endpoint of the screw
, with respect to the principal frame, in which
P with the coordinates
has the following parametric equations in terms of the coefficients
and
:
In the above equations, dividing equation
x by equation
y and equation
z, respectively, and rearranging, yields
Substituting the above expressions of
and
into equation y of Equation (5) and rearranging yields
The above equation is simply the expression of the surface
proposed by Hunt [
28]. For an insight into the inner structure of this surface, it is plotted in
Figure 1 by Equation (5) when we set
,
, and
, and set the coefficients
and
as the two primary variables.
The curves in
Figure 1(A1–A3) are the contour lines of the surface
perpendicular to the
z-,
y-, and
x-axes of the principal frame, respectively. The primary variables
and
both continuously take values in the interval
. The surface
grows up from a patch of surface to a semi-closed surface with four cavity structures, as shown in
Figure 1(C3,C2,C1,B3,B2,B1,A3) when the range parameter
d takes the values 0.5, 1, 1.5, 2, 3, 8, and 30 in turn, respectively. When the pair of primary variables
takes the pair of values
and
, the coordinate
y in Equation (5) obtains its two extreme values
and
, respectively. Similarly, coordinate
z in Equation (5) obtains the two extreme values
and
when the pair of primary variables
takes the pair of values
and
, respectively. So, the cavity structures begin to appear when
d = 1, as shown in the
Figure 1(C2). In particular, since coordinate
x in Equation (5) has no extreme values, the four cavity structures infinitely approach the
x-axis of the principal frame, but will never be close even though the primary variables
and
both tend to positive or negative infinity. The surface
describes the exact distribution of the position of the
screw axes in Euclidean three-space.
3. Densest Distribution Zone of Screw Axes
In this section, we will derive the expression of the finite distribution zone and then reveal that the position vector endpoint is the particular case of the striction points on the axis of .
When two lines lying on a ruled surface infinitely approach each other, the two endpoints of their common perpendicular segment infinitely tend to a limit point. This limit point is called a striction point of the ruled surface. Analogously, every one of the screw axes of a general three-system will get striction points when it is infinitely close to its surrounding screw axes in all directions in its infinitesimal neighborhood space. So, the striction points of the general three-system constitute a finite striction zone. According to its definition, the screw axes all pass through this zone, and they most densely distribute in this zone. Therefore, we define this finite striction zone as the densest distribution zone of the general three-system. In the following lines, we will derive the parametric equation of the densest distribution zone.
Setting
as the position vector of the screw
, its axis has the following Plücker coordinates:
where
is the unit vector. Without a loss of generality, the axis of another screw,
, which is just in the infinitesimal neighborhood space of the screw
, is denoted by:
The relative location between
and
is shown in
Figure 2.
As seen in
Figure 2,
is the screw axes’ common perpendicular segment.
P and
are their respective position vector endpoints, and
is a striction point generated by
and
when they are infinitely close to each other. According to Equations (1–31) of [
29], the point
has the form
Considering
,
,
, and
are the vector functions of the two variables in
and
, we set
where
and
are the increments of the variables
and
, respectively. According to Taylor’s formula for functions of two variables, we have
where
and
are the remainder vectors of
and
, respectively, which both consist of higher-order vector derivatives and mixed derivatives, and their coefficients are all the infinitesimal amounts of more higher-order than
and
when they infinitely tend to zero. By setting
Equation (12) can be rewritten as
Substituting Equation (14) into Equation (10) yields
In Euclidean two-space endowed with the Cartesian coordinate system
, when point
infinitely tends to point
, i.e.,
and
both get near to zero,
moves along the moving segment
toward
and then it moves along
toward
, i.e., point
infinitely tends to point
and then tends to striction point
in the principal frame of the general three-system. Consequently, taking the limitation of Equation (15) when
and
both infinitely get near to zero, we have:
provided the limit exists, where
In particular, we introduce the function
, which represents an arbitrary curve in
plane of the Cartesian coordinate system
, for that the point
can arbitrarily tend to the fixed point
when
and
both get near to zero. Correspondingly,
can be arbitrarily and infinitely close to
in its infinitesimal neighborhood space in the principal frame of the general three-system. Substituting Equation (17) into Equation (16) will degenerate it into the limit of the vector function of one variable as follows:
Since
is a unit vector, we have
. So, the above equation reduces to
For the compactness of the following formulas, we set
,
,
, and
without a loss of generality. Although Equation (19) is just the formula of the striction point of ruled surfaces, it can be used to derive the
striction points of the general three-system. Substituting
,
,
, and
k into Equation (19) and rearranging lead to
where
is defined as the distribution coefficient of the
striction points within the general three-system because it involves the three primary variables
,
, and
k, and the two parametric variables
e and
g. In order to analyze the distribution range of the
striction points on the axis of
, we set
,
,
e, and
g as the parametric variables and
k as the primary variable taking values in the interval
. According to Equation (21), when the point
tends to the point
in different directions, i.e.,
k takes different values, we will get different striction points on the screw axis
, i.e., the limit of Equation (16) does not exist in general cases. Setting
yields
This means that there exist two directions along which the screw is infinitely close to its infinitesimal neighboring screws, generating two limit points which are its position vector endpoints. In particular, when and , substituting them into Equation (21) and setting yields . When and , substituting them into Equation (21) and setting yields . When and , substituting them into Equation (21) and setting yields or .
Remark: The screwsare defined as the singular screws which always receive a unique striction point (i.e., their position vector endpoint), whatever values k takes. So, the limit of Equation (16) exists for these singular screws.
When
attains its two extreme values
and
as follows:
where
stands for
and
, which are associated with the symbol “
” and the symbol “
” in the symbol “
”, respectively. Setting
and
, and setting
and
as the primary variables, we can get the surface of
and
in the Cartesian coordinate system
, as shown in
Figure 3.
As seen in
Figure 3, the curves on the surfaces are their contour lines. Equations (24) and (20) illustrate that when
k takes values in the interval
,
striction points on every screw axis of the general three-system symmetrically distribute on both sides of the position vector endpoint
P, as shown in
Figure 4.
As seen in
Figure 4,
and
are defined as the top- and bottom-end striction points, respectively. So far, we can formulate the densest distribution zone of the general three-system as:
where
is a three-dimensional vector function of three variables in
,
and
v. The variable
v partitions the densest distribution zone
into
striction point surfaces. Among these surfaces, there are three important surfaces. When
,
holds. We define the position vector endpoint
P as the centric striction point. So, the parametric surface
is also called the centric striction point surface (the surface
). Clearly, the position vector endpoint
P is a particular case of the striction points. When
,
We define the parametric surface
as the top-end striction point surface. When
,
We define the parametric surface
as the bottom-end striction point surface. The above two parametric surfaces constitute the boundary surface of the densest distribution zone. Their parametric equations have the form
where
stands for
and
, which are associated with the symbol “
” and the symbol “
”, respectively, in the symbol “
”.
By the procedure of the surface
evolving into the boundary surface of the densest distribution zone, we have an insight into its inner structure. In Equation (25), setting
and
and setting the primary variables
and
, both continuously taking values in the interval
yield the evolving procedure as shown in the
Figure 5(E1–E3,D1–D3,C1–C3,B1–B3,A1–A3), when
v takes the values 0, ±0.3, ±0.5, ±0.8, and ±1, respectively. The curves on the surfaces are their contour lines.
To compare the relative location of the five evolutionary surfaces in
Figure 5, we set the primary variables
and
to take values in the interval
and the range parameter
d to take the values 0.3, 0.5, 0.8, 1, 1.3, 1.6, 2, 2.5, 3, 6, 12, and 24 in turn, respectively. The growing up process of the five evolutionary surfaces is shown in
Figure 6.
From
Figure 6, we know that the evolutionary surface associated with a bigger absolute value of
v (in blue) enfolds the one associated with a smaller absolute value of
v (in blue). In other words, top- and bottom-end striction point surfaces are indeed the boundary of the densest distribution zone. From the above discussion follows at once an important
proposition: the striction curve of the ruled surface, which consists of arbitrary
screw axes of the general three-system, must be located inside the densest distribution zone.
Figure 6 describes the inner structure of the densest distribution zone in detail.
The boundary surface of the densest distribution zone of the general three-system is our original contribution. Its beauteous details are shown in
Figure 7 by four close-up views.
Figure 7A,B are the top-end striction point surfaces.
Figure 7C,D are the bottom-end striction point surfaces.
4. Distribution Regularity of the Two-Subsystems of the General Three-System
The above two sections discuss the densest distribution zone of the screw axes of the general three-system in Euclidean three-space and decompose this zone into striction point surfaces by the primary variable v in Equation (25) to explore its inner structure. In this section, we attempt to establish the relationship between this zone and the cylindroids.
Equation (2) maps the
screws within a general three-system to the
points on the
plane. These points can be partitioned into
straight lines (a family of straight lines). Clearly, every straight line corresponds to a two-subsystem [
6] of the general three-system, whose
screw axes lie on a two-subsystem cylindroid (TC), as shown in
Figure 8. In other words, the distribution zone of the general three-system can be partitioned into
cylindroids.
Remark: A partitioning method corresponds to a family of two-subsystems which consists oftwo-subsystems, and then to a family of straight lines in theplane. Any family of straight lines in theplane can be expressed aswhere the slope k, primary variable, and intercept c are real numbers and take values in the interval. So, the parameter k essentially determines thepartitioning methods of the general three-system intotwo-subsystems. When the partition method has been given, the parameter c taking values in the intervalwill generate the two-subsystems of this family of two-subsystems. In a two-subsystem, the parametertaking values in the intervalwill generate thescrews. In order to decompose the densest distribution zone into cylindroids, we establish the relationship between the cylindroid and the three parameters , c, and k in theorems 1, 2, and 3, respectively. In Theorem 1, k and c are set as the parametric variables and is set as the primary variable.
Theorem 1. On a certain cylindroid lie thescrews of a general three-system associated with thepoints on any straight line of theplane, while on two planar pencils lie those associated with thepoints on the two exceptional straight linesand, respectively.
Proof. Substituting Equation (29) into Equation (2) and rearranging lead to the unit Plücker coordinate of any screw within this two-subsystem, as follows:
When the primary variable is
, the screw axis of
traces a ruled surface whose striction curve has the following parametric equation:
From Equation (31), we know that when
The striction curve
attains its two extreme endpoints
and
, respectively, in the principal frame, and
stands for
and
, which are associated with the symbol “
” and the symbol “
” in the symbol “
”, respectively. Setting the length of the secant vector
of the striction curve
as
A, we have
Setting
and
, and solving the above equation for
c lead to
. This means the striction curves of the ruled surfaces associated with the lines
degenerate into the two points
, respectively. Further computing finds that the screws
and the screw
have the same pitch
, and they constitute the basis screws of the first special two-systems [
4]. In other words, the screws associated with lines
form two first special two-systems, respectively. So, the screws associated with the exceptional lines
lie on two planar pencils centered at points
, respectively.
Setting
and
and solving Equation (33) for
leads toand solving Equation (33) for
and
The above two solutions both contradict with
k2 > 0. That means there are no more straight lines in the
plane associated with the ruled surfaces satisfying the condition
. When
, the straight line determined by the unit vector
of the secant vector
and the point
has the form
Since all points in Equation (31) satisfy the above equation, the striction curve
is a straight line. In addition, the two screws
passing through the two extreme endpoints
and
, respectively, are mutually perpendicular and are both perpendicular to the striction line
. When
the striction line
attains the midpoint
of the two extreme endpoints
and
twice, and
stands for
and
, which are associated with the symbol “
” and the symbol “
” in the symbol “
”, respectively. In particular, if
k and
c make the denominator
of
equal to zero, then
and
. Furthermore, the two screws
and the striction line
are mutually and orthogonally concurrent at the midpoint
of the striction line
. Clearly,
are two principal screws and
are two end screws [
4], respectively. To summarize, the ruled surface traced by
possesses all the geometric features of a cylindroid. Therefore, Theorem 1 is proven. The cylindroid constructed in Theorem 1 is defined as the two-subsystem cylindroid (TC). □
In Theorem 2, k is set as a parametric variable and and c are set as the primary variables. The lines with the same slope k in the plane correspond to a certain family of two-subsystems.
Theorem 2. On a certain cylindroid lie thespine lines associated with thetwo-subsystem cylindroids within any family of two-subsystems of a general three-system.
Proof. The unit Plücker coordinate of the spine line of a two-subsystem cylindroid has the form
where the superscript
s represents the spine line. When
c takes values in the interval
,
traces a spine-ruled surface in the principal frame whose striction curve has the form
When
, the striction curve
attains its two extreme endpoints
and
as follows:
where
stands for
and
, which are associated with the symbol “
” and the symbol “
” in the symbol “
”, respectively, and are associated with the symbol “
” and the symbol “
” in the symbol “
”, respectively. Setting the length of the secant vector
of the striction curve
as
A leads to
Clearly,
holds for any value of
k. The straight line determined by the unit vector
of the secant vector
and the point
has the form
Since all points in Equation (39) satisfy the above equation, the striction curve of the spine-ruled surface is a straight line. In addition, the two screws passing through the two extreme endpoints and , respectively, are mutually perpendicular and are both perpendicular to the striction line . The striction points are both the midpoints of the two extreme endpoints . The line vectors and have the direction vectors and , respectively. Moreover, these two line vectors, and , and the striction line are mutually and orthogonally concurrent at the midpoint of the striction line . Clearly, and are two principal screws, and are two end screws, respectively. To summarize, the spine-ruled surface traced by possesses all the geometric features of a cylindroid. Therefore, Theorem 2 is proven. The cylindroid constructed in Theorem 2 is defined as the first-order spine cylindroid (FSC). □
In Theorem 3, k, , and c are set as the primary variables. We will get the families of two-subsystems within a general three-system.
Theorem 3. On a certain cylindroid lie thespine lines associated with thefirst-order spine cylindroids within a general three-system.
Proof. The unit Plücker coordinate of the spine line of a first-order spine cylindroid has the form
When
k takes values in the interval
,
traces another spine-ruled surface whose striction curve has the form
When
, the striction curve
attains its two extreme endpoints
and
as follows:
where
stands for
and
, which are associated with the symbol “
” and the symbol “
” in the symbol “
”, respectively. Clearly, the length of the secant vector
is
e, which is greater than zero forever. The straight line determined by the unit vector
of the secant vector
and the point
has the form
Since all points in Equation (44) satisfy the above equation, the striction curve of the spine-ruled surface is a straight line. In addition, the two screws passing through the two extreme endpoints and , respectively, are perpendicular mutually and are both perpendicular to the striction line . The striction points are the midpoints of the two extreme endpoints . The line vectors and have the direction vector and , respectively. So, the two line vectors, and , and the striction line are mutually and orthogonally concurrent at the midpoint of the striction line . Clearly, and are the two principal screws and are the two end screws. To summarize, the spine-ruled surface traced by possesses all the geometric features of a cylindroid. Therefore, Theorem 3 is proven. The cylindroid constructed in Theorem 3 is defined as the second-order spine cylindroid (SSC). □
The above three theorems reveal the principle of constructing the densest distribution zone of
screw axes of the general three-system by cylindroids of the general two-system in Euclidean three-space, as shown in
Figure 9.
The densest distribution zone of the general three-system can be decomposed into
TCs, whose spine lines constitute an FSC. There are actually
decomposition methods corresponding to
FSCs, whose spine lines constitute the SSC. The above decomposition procedure is shown in
Figure 9 in detail.
Setting
,
, and
,
Figure 10 depicts the relative locations among TCs, FSCs, and SSC in the densest distribution zone of the general three-system.
Figure 10(A1) shows the TC corresponding to the two-subsystem associated with the line
in the
plane. The length of the segment lying on the TC represents twice the pitch of the screw passing through the line.
Figure 10(A2) shows the FSC corresponding to the family of two-subsystems associated with the family of straight lines
in the
plane. The segments lying on the FSC are the spine lines of the TCs within the above family of two-subsystems. The spine lines of the two TCs (
) are coincident with the two end screw axes of the FSC, respectively.
Figure 10(A3) shows the SSC (in red curves and lines). The red segments lying on the SSC are the spine lines of the FSCs corresponding to all the families of two-subsystems within the general three-system (
,
, and
). The spine line of the FSC (
) is coincident with the principal screw axis of the SSC.
Figure 10(B1) shows the SSC (in red curves and lines) and two FSCs (
) whose spine lines are coincident with the two end screw axes of the SSC, respectively. The three cylindroids are represented by boundary curves and segments lying on the surface.
Figure 10(B2,B3) show that the boundary surface of the densest distribution zone completely wraps the three cylindroids shown in
Figure 10(B1). So, the
proposition in the end of
Section 3 is verified again geometrically.
Hunt and Phillips [
5,
28] presented a distribution characteristic about the central plane of the two-subsystem cylindroid (see Theorem 4). In this section, we try to prove it with a simple computation.
Theorem 4. Through the origin of the principal frame passes every central plane of TC (including degenerated planar pencils) within a general three-system.
Proof. Substituting the straight line
into Equation (2) leads to the Plücker coordinate of any screw within a two-subsystem, as follows:
When
and
, the above equation reduces to
Clearly, the pitch
of
is equal to
for any value of
. So, the screws represented by Equation (48) belong to the first special two-system. Since these screws lie on a concentric planar pencil, the planar pencil is coincident with the central plane of this two-subsystem. Without a loss of generality, we take two screws as follows:
and then the normal vector of the central plane has the form
In addition, the equation of the central plane [
29] has the form
where
is the position vector endpoint of
. The origin
of the principal frame satisfies the above equation of the central plane. When the screw axes of a two-subsystem lie on a cylindroid, the pitch
hi of any screw has the form
Differentiating the above equation with respect to
leads to
Setting
, we receive the two stationary points
where
stands for
and
, which are associated with the symbol “
” and the symbol “
” in the symbol “
”, respectively. Substituting them into Equation (47) yields the two principal screws
and
(see the ordinary case of
Appendix A). In particular, if k and c make the denominator
of
equal to zero, then
and
. For the above three particular cases, three pairs of principal screws are listed in the particular cases of
Appendix A. The central plane determined by the above four pairs of principal screws has the form
where the normal vector
,
is the position vector endpoint of
. The origin
of the principal frame satisfies the above equation. Therefore, Theorem 4 is proven. □
The above four theorems exactly describe the geometrical regularity of the distribution of all two-subsystems in Euclidean three-space. They will provide exact guides for the axode planning of three degree-of-freedom parallel mechanisms in its reference frame.
5. Varying-Pitch Ruled Surface
The distribution zone of the general three-system can also be decomposed into pitch-hyperboloids. Furthermore, cylindroids and pitch-hyperboloids can be regarded as two particular decomposition methods of the distribution zone of the general three-system. In this section, we will propose a generalized decomposition method: varying-pitch ruled surface based on the pitch-hyperboloid.
It is well known that screws with the same pitch value lie on the same pitch-hyperboloid [
4,
5]. According to the pitch value, the distribution space of the
screw axes within a general three-system can be partitioned into
pitch-hyperboloids, two concentric planar pencils, and two coordinate axes of the principal frame. Setting the pitch
as a parametric variable, Equation (3) can be rewritten as
When , Equation (56) degenerates into two lines . Considering the proof of Theorem 1, we know that screws with pitch lie on two concentric planar pencils. When , Equation (56) degenerates into the origin of the plane, i.e., the screw with pitch lies on the x-axis of the principal frame. When , Equation (56) does not hold, but the screw with pitch lies on the z-axis.
When
,
and
hold. Equation (56) can be rewritten as the stand equation of the family of ellipses as follows:
Their function expressions have the form
When
,
and
hold. Equation (56) can be rewritten as the stand equation of the family of hyperbolas as follows:
Their function expressions have the form
Setting
,
, and
, the curves represented by Equation (56) are plotted in the
plane when the parametric variable
takes values in the interval
, as shown in
Figure 11.
In
Figure 11, the top endpoint a of the major axis of the hyperbola is at
. The intersection b of the straight line
and the
-axis is at
. When
infinitely approaches
, point a infinitely tends to b, and the slope of the asymptotes of the hyperbola infinitely tends to zero. That means the top half of the hyperbola gradually becomes the line
. When
infinitely approaches
, point a infinitely tends to the positive infinity of the
-axis. That means the regions above the line
and below the line
can be completely covered by the family of hyperbolas. The top endpoint c of the ellipse’s minor axis is at
. The right endpoint d of the ellipse’s major axis is at
. When
infinitely approaches
, points c and d both infinitely tend to the origin of the
plane. When
infinitely approaches
, point c infinitely tends to point b, and point d infinitely tends to the positive infinity of the
-axis. That means the region between the two lines
can be completely covered by the family of ellipses. To summarize, the
points of the
plane can be completely covered by the curves represented by Equation (56) when
continuously takes values in the interval
.
Substituting Equation (58) and Equation (60) into Equation (20) yields the parametric equation of the striction curve of the ruled surface on which lie screws with the pitches
and
, respectively, as follows:
where
and
are associated with the symbol “
” and the symbol “
” in the symbol “
”, respectively, and are associated with the symbol “
” and the symbol “
” in the symbol “
”, respectively.
Setting
,
, and
, when
, the corresponding ruled surface, striction curve (in red), and pitch-hyperboloid are as shown in
Figure 12A. When
, the corresponding ruled surface, striction curve (in red), and pitch-hyperboloid are as shown in
Figure 12B.
Remark: Although the ruled surfaces lie on the pitch-hyperboloids, their central ellipses do not cover the striction curves of the ruled surfaces. In the following lines, variation of the parameter in Equation (56) yields the varying-pitch ruled surface.
In particular, if the parametric variable
in Equation (56) is substituted by a continuous function of
, i.e.,
, Equation (56) can be rewritten as
Substituting the function expression of Equation (62) into Equation (2) leads to a serial of screws whose pitch value varies according to any desired continuous function
. These screws lie on a ruled surface which is defined as the varying-pitch ruled surface. For example, setting
, and considering
, the function expression of Equation (62) has the form
Setting
, and considering
, the function expression of Equation (62) has the form
Setting
,
, and
, and substituting Equation (63) and Equation (64) into Equation (2) yield two serials of screws, respectively. Their varying-pitch ruled surfaces and striction curves (in red) are plotted in the
Figure 13A and
Figure 13B, respectively. The parametric variable c takes the values 3 and 2, as seen in the
Figure 13A and
Figure 13B, respectively.
6. Numerical Examples of the 3-RPS Pyramid Mechanism
In axode planning, there is a basic problem: given an axode arbitrarily in the reference frame of a parallel mechanism, how can one visually tell whether this axode can be produced by the mechanism when it passes through a serial of configurations? The necessary condition is that its striction curve is inside the zone consisting of a serial of densest distribution zones associated with a serial of configurations, respectively.
Huang [
30] pointed out that the twist screw system of the 3-RPS pyramid mechanism is just a general three-system of screws. So, in this section, we present a numerical example of a 3-RPS pyramid mechanism to demonstrate that the necessary conduction is valid. The mechanism consists of a base
, a moving platform
, and three identical-limb RPSs. Here, R represents the revolute joint, P denotes the prismatic joint, and S stands for the spherical joint. In particular, the axis of the R joint is perpendicular to that of the P joint. The reference frame
attached to the base has its origin located at the point
and its three axes are along the three ones of the R joints of the base, respectively. The body frame
attached to the moving platform has its origin located at the midpoint of the triangle
, its
z-axis perpendicular to the plane
, its
y-axis parallel to the segment
and in the plane
, and its
x-axis defined according to the right-hand rule as shown in
Figure 14. In the configuration shown in
Figure 14, the lengths of three P joints are M, and the mechanism takes on the form of a cube.
Limb
i imposes a constraint wrench screw
on the moving platform. They pass through the three centers of the S joints and are parallel to the three axes of three R joints, respectively. So, they are linearly independent and constrain three translational freedoms of this mechanism, whatever configuration it is under. The reciprocal screw system of the above three constraint wrench screws is a general three-system of screws and stands for three rotational freedoms of the mechanism. The 3-RPS pyramid mechanism sets its structural parameters and input parameters as follows:
where
represents the initial orientation
of the body frame with respect to the reference frame. M is the length of the segment
.
,
and
are the input parameters of the three P joints, respectively, when the mechanism performs a motion task. In this motion task, a serial of densest distribution zones and an axode can be obtained as shown in
Figure 15. The
Figure 15(A1–A3) display the densest distribution zones from three different views, respectively. The striction curve (in blue) of the axode is shown in the
Figure 15(B1–B3) from three different views, respectively. As shown in
Figure 15, the striction curve indeed is located inside the zone consisting of a serial of densest distribution zones.