MSMechanical SciencesMSMech. Sci.2191-916XCopernicus PublicationsGöttingen, Germany10.5194/ms-9-297-2018Solution Region Synthesis Methodology of RCCC Linkages for Four PosesSolution Region Synthesis MethodologyHanJianyoujyhan@ustb.edu.cnCaoYangSchool of Mechanical Engineering, University of Science and
Technology Beijing, Beijing 100083, ChinaJianyou Han (jyhan@ustb.edu.cn)24September20189229730528December201730July20188September2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://ms.copernicus.org/articles/9/297/2018/ms-9-297-2018.htmlThe full text article is available as a PDF file from https://ms.copernicus.org/articles/9/297/2018/ms-9-297-2018.pdf
This paper presents a synthesis methodology of RCCC linkages based
on the solution region methodology, R denoting a revolute joint and C
denoting a cylindrical joint. The RCCC linkage is usually synthesized via its
two defining dyads, RC and CC. For the four poses problem, there are double
infinite solutions of the CC dyad, but there is no solution for the RC dyad.
However, if a condition is imposed that leads to a coupling of the two dyads,
a maximum of four poses can be visited with the RCCC linkage. Unfortunately,
until now, there is no methodology to synthesize the RCCC linkage for four
given poses besides optimization method. According to the coupling condition
above, infinite exact solutions of RCCC linkages can be obtained. For
displaying these RCCC linkages, we first build a spherical 4R linkage
solution region. Then solutions with circuit and branch defects can be
eliminated on this solution region, so that the feasible solution region is
obtained. An RCCC linkage can be obtained by using the prescribed spatial
positions and selected a value on the feasible solution region. We take
values on the feasible solution region by a certain step length and many
exact solutions for RCCC linkages can be obtained. Finally we display these
solutions on a map, this map is the solution region for RCCC linkages.
Introduction
The synthesis of spatial RCCC linkages has been received much more attentions
recently. A formulation based on dual algebra is proposed for the approximate
synthesis of RCCC linkage motion generation (Angeles, 2014). An approach is
introduced for the synthesis of the axode of an RCCC linkage (Figliolini et
al., 2016). An RCCC linkage is synthesized by a robust optimization method
(Al-Widyan and Angeles, 2012). Fourier series theory is used for path
generation of RCCC linkage (Sun et al., 2012, 2017). A semi-graphical
approach is proposed for the synthesis of 4C linkage for five given poses to
obtain a robust solution (Bai and Angeles, 2012). A complete classification
scheme is developed for planar, spherical 4R and spatial RCCC mechanisms
(Murray and Larochelle, 1998). Larochelle addressed the issue of branch and
circuit analysis of spatial 4C mechanisms for rigid body guidance
(Larochelle, 2000).
A CC dyad.
There are no exact solutions for the synthesis of RCCC linkage for four
prescribed coupler poses (Bai and Angeles, 2012). In some literatures (Bai
and Angeles, 2015), RC dyad is synthesized by optimization method and CC dyad
is synthesized by line congruence, only one approximate solution for RCCC
linkage can be obtained for four poses. In this paper, we add constraints
between RC dyad and CC dyad, thus the number of unknowns more than the number
of equations and infinite exact solutions for RCCC linkage are obtained. For
synthesizing more RCCC linkages, we propose a solution region methodology by
which infinite exact solutions can be obtained. The synthesis process of this
methodology is separated into two parts. The first part is to build a
spherical 4R linkage solution region based on Burmester curves. The Burmester
curves are determined by four orientations from the four poses. Then we
eliminate those solutions having circuit and branch defects and obtain the
feasible solution region. In second part, we couple the constraint conditions
for an RC dyad and a CC dyad, and obtain 23 equations with 24 unknowns.
Therefore, there is one parameter to be chosen freely. Finally we select
different values on the feasible solution region, and solve these equations
using Bertini software, and build an RCCC linkage solution region for
displaying these solutions. The solution region methodology introduced in
this paper has been applied in the synthesis of planar and spherical linkages
(Han and Qian, 2009; Yang et al., 2011, 2012; Yin et al., 2012; Cui and Han,
2016). We also synthesized the 5-SS platform linkage by solution region
methodology (Han and Cui, 2017). We proposed the primary idea of the
synthesis of RCCC linkages for four poses by solution region map (Cao and
Han, 2016; Han and Cao, 2017), but the method of building solution region
isn't given.
An RC dyad.
Problem formulation
A spatial CC dyad is shown in Fig. 1. The CC dyad is meant to carry the rigid
body through a set of given poses, specified by coordinates pi
(pix, piy, piz) and orientations (θi, αi,
βi) respect to a reference coordinate frame (xyz) as shown in
Fig. 1, where orientation angles θi, αi, βi are
three Euler angles. The unit vectors u1 and v are
along the CC dyad cylindrical pairs moving and fixed axes, respectively.
Vector ui denotes the unit vector of u1 at the
ith pose. The CC dyad can be geometrically regarded as a link composed of
two skew lines, jointed each other by means of a third line, their common
perpendicular (Fig. 1). M1 is the intersection point of the common
perpendicular and moving axis. F1 is the intersection point of the
common perpendicular and fixed axis. h donates the distance between the
moving and the fixed axes of the CC dyad. S denotes the sliding
displacement of the cylindrical pair from the first to the ith pose, S is
defined as
S=∑i=2nSi
where Si denotes the sliding displacement of the cylindrical pair from
the pose i-1 to the pose i. When the sliding direction of the cylindrical
pair (while the coupler moves from one pose to the next) is in the same
direction with vector v, Si is positive, whereas negative. Note
that the CC dyad must satisfy two constraints. The angle of twist between the
fixed axis and the moving axis must remain constant from 1st pose to ith
pose. The moment of vector ui about axis v must remain
constant from 1st pose to ith pose. Hence, the CC dyad constraint equations
can be written as follows.
uiTv=u1Tvi=2,3,…,nvTmi-fi×ui=vTm1-f1×u1(i=2,3,…,n)
In Eq. (2), m1, mi, f1 and
fi are the position vectors of M1, Mi, F1 and
Fi respectively. Vectors u1, ui and
v are unit vectors, Therefore
u12=1,v2=1
An RCCC linkage.
For ensuring the vector m1-f1 is orthogonal to
their unit vectors, we have
u1Tm1-f1=0,vTm1-f1=0
In Eq. (1), ui can be expressed as
ui=Riu1(i=2,3,…,n)
In Eq. (5), Ri is a rotation matrix which rotates rigid body
from 1st to ith orientation. For rotating rigid body from 1st to ith
orientation, first, we rotate the rigid body to make its orientation coincide
with the reference coordinate by three Euler angles (-β1,
-α1 and -θ1 around the z-, x- and z-axes,
respectively), then rotate it to the ith by three Euler angles (θi, αi and βi around the z-, x- and z-axes,
respectively). Therefore Ri can be expressed as
Ri=ZβiXαiZθiZ-θ1X-α1Z-β1(i=2,3,…,n)
where X, Z are pure rotation matrixes around x-,
z-axes, respectively. All rotation matrixes follow the right-hand rule.
Spherical Burmester center and circle point curves.
mi and fi can be expressed as matrix
forms
Uixmi=RiU1xm1+Pixm1(i=2,3,…,n)Vxfi=Vxf1(i=2,3,…,n)
where U1x, Uix, Vx and Pix are cross-product matrixes of vectors
u1, ui, v, and pi.
Burmester circle and center point curves in the α-β
plane.
Equations (1) to (4) constitute a set of 2n+2 (2(n-1)+4=2n+2) constraint
equations with twelve unknowns, which are scalar components of u1, v, m1 and f1. Therefore, the maximum
number of poses, which can be specified for a CC dyad to be used for rigid
body guidance, is five.
The RC dyad shown in Fig. 2 must satisfy all of the constraints imposed by a
CC dyad plus an additional displacement constraint, the sliding Si
along the fixed axis is zero
Si=fi-fi-1=0(i=2,3,…,n)
Equations (1), (2), (3), (4) and (9) constitute a set of 3n+1
(3(n-1)+4=3n+1) constraint equations with twelve unknowns, which are scalar
components of u1, v, m1 and
f1. Therefore, the maximum number of poses, which can be
specified for a CC dyad to be used for rigid body guidance, is three with
arbitrary choice of any two scalars.
An RCCC linkage is shown in Fig.3. We use u1, v,
m1 and f1 to represent the RC dyad and use
u1∗, v∗, m1∗ and
f1∗ to represent the CC dyad. If we synthesize RC dyad
and CC dyad separately, then simply couple the RC dyad and CC dyad with a
coupler link for a RCCC linkage to be used for rigid body guidance, the
maximum number of specified poses is three. When we add a constraint between
two dyads instead of Eq. (9), we have
fi-g1=f1-g1(i=2,3…,n)
where g1 is the position vector of G1 as shown in Fig. 3.
g1 can be expressed by vectors v, f1,
v∗ and f1∗. Then the RCCC linkage
synthesis problem involves 5n+3 (5(n-1)+8=5n+3) equations, Eqs. (1), (2),
(3), (4) and (10). These equations are to be solved for 24 unknowns, which
are components of the eight vectors u1, v,
m1, f1, u1∗,
v∗, m1∗ and f1∗.
The maximum number of poses, which can be specified for an RCCC linkage to be
used for rigid body guidance, is four with arbitrary choice of one scalar.
Spherical 4R linkage solution regions.
Process of synthesizing RCCC linkages
We separate the synthesis process into two parts. For determining the range
of αc (αc is arbitrary choice from
24 unknowns, here αc is defined as the latitude of a point
on spherical circle curve which will be mentioned below), we build a
spherical 4R linkage feasible solution region. Then we set up the synthesis
formulation for synthesizing RCCC linkages.
Determination of spherical Burmester curves and Building of the
spherical 4R linkage solution regions
In this section, we determine the Burmester curves and build spherical 4R
linkage solution region by these curves. On this solution region, we can
classify the mechanism types and eliminate circuit and branch defects.
RCCC solutions corresponding the parts on the feasible spherical 4R
linkage solution region.
Spherical Burmester curves
Unit vector v is along the fixed axis, whereas unit vector
u1 is along the moving axis. For the four given poses problem,
n=4.
Let
B=u1TR2-ITu1TR3-ITu1TR4-ITT
then Eq. (1) can be rewritten as
Bv=0
Since unit vector v cannot be zero, B must be non-full
rank. Therefore, its determinant is zero, i.e., B=0. The spherical Burmester circle point curve can be expressed as
B=0u12=1
Likewise, the spherical Burmester center point curve can be expressed as
A=0v2=1
where A=vTR2T-IT,vTR3T-IT,vTR4T-ITT (Bai and
Angeles, 2012).
The solution region of RCCC linkages.
We use spherical coordinates on the unit sphere, i.e., latitude and
longitude, to describe the unit vectors of all four cylindrical directions.
Let αc and βc be the longitude and
latitude of a point on spherical circle curve, and αo and
βo be the latitude and longitude of a point on spherical center
curve.
Therefore
u1=u1xu1yu1z=cosβccosαccosβcsinαcsinβc,v=vxvyvz=cosβocosαocosβosinαosinβo
Each point on the Burmester circle curve corresponds to the point on the
Burmester center curve. Connecting the corresponding points, we obtain an RR
dyad. A spherical 4R linkage can be synthesized by selecting two different
points on the Burmester circle (or center curve).
Building of the spherical solution regions
Let αc (the latitude of Burmester circle point curve) as the
x- and y-axes, we build a solution region for displaying infinite
spherical 4R linkage solutions, as shown in Fig. 6a. Building process in
detail is shown in Sect. 4. The circuit and branch defects can be eliminated
on this solution region. The feasible solution region is built after
eliminating the defects (Larochelle, 2000) from spherical 4R solution region,
as shown in Fig. 6b.
Four configurations of the RCCC linkage for point p1.
Four configurations of the RCCC linkage for point p2.
The synthesis formulation of RCCC linkages
We have discussed the number of the unknowns and the equations for
synthesizing RCCC linkages for four poses above. For synthesizing driving
link, substituting Eq. (5) into Eq. (1), we have
u1TRiT-Iv=0(i=2,3,4)
Substituting Eqs. (7) and (8) into Eq. (2), we have
u1TPixRiTv+u1TVxRiT-Vxf1+m1TU1xTRiT-U1xTv=0(i=2,3,4)
Because the Eqs. (16) and (17) can apply for both CC and RC dyad, we use
u1, v, m1, f1 to represent the
direction and position vectors for RC dyad (driving link); we use
u1∗, v∗, m1∗,
f1∗ to represent the direction and position vectors for CC
dyad (driven link). The two sets of vectors can be applied for Eqs. (16) and
(17). Therefore, substituting the unit vectors and position vectors of driven
link into the Eqs. (16) and (17), we have
u1∗TRiT-Iv∗=0(i=2,3,4)u1∗TPixRiTv∗+u1∗TVxRiT-Vxf1∗+m1∗TU1xTRiT-U1xTv∗=0(i=2,3,4)
The twelve equations are list below
u1TRiT-Iv=0(i=2,3,4)u1∗TRiT-Iv∗=0(i=2,3,4)u1TPixRiTv+u1TVxRiT-Vxf1+m1TU1xTRiT-U1xTv=0(i=2,3,4)u1∗TPixRiTv∗+u1∗TVxRiT-Vxf1∗+m1∗TU1xTRiT-U1xTv∗=0(i=2,3,4)
Equations (3) and (4) are unit vectors equations and orthogonality equations
(Suh and Radcliffe, 1978) for driving link. We rewrite the Eqs. (3) and (4)
with unknowns u1∗, v∗,
m1∗ and f1∗ to represent the unit
vectors equations and orthogonality equations for driven link
u1∗Tm1∗-f1∗=0,v∗Tm1∗-f1∗=0u1∗2=1,v∗2=1
Equation (10) constrains the sliding displacement between the driving link
and the fixed link. So, there are 23 equations (Eqs. 3, 4, 10, 20 and 21) to
solve 24 unknowns, i.e., u1, v, m1,
f1, u1∗, v∗,
m1∗ and f1∗. We have free choice of
one from 24 unknowns.
In this paper, we select αc on x-axis of spherical 4R
linkage feasible solution region as the arbitrary choice scalar. Then
23 equations can be solved for other 23 unknowns using the Bertini software
(Bates et al., 2006).
In Table 1, αi, βi, θi are the Euler
angles and pi (pix, piy, piz) are the spatial
coordinates. The design demands are:
Linkage type: crank rocker;
The ranges of x-axis of points M1 and M1∗ are restricted in
(-100, 100);
No circuit and branch defects.
Substituting four orientations (Table 1) into Eqs. (13) and (14), the
spherical Burmester circle and center point curves are obtained, as shown in
Fig. 4. Substituting u1 in Eq. (15) into Eq. (13), and substituting
v in Eq. (15) into (14), then Eq. (13) has only two variables
αc and βc; Then Eq. (14) has only two
variables αo and βo; Therefore, we convert the spatial
Burmester curves into the α-β plane, as shown in Fig. 5.
Let αc (the latitude of Burmester circle point curve) as the
x- and y-axes, we build a solution region for displaying infinite
spherical 4R linkage solutions, as shown in Fig. 6. We classified the
solution region with nine parts with different colors. In the white region,
there is no linkage. The rest eight parts are classified by eight types of
mechanisms as listed in Fig. 6. Circuit and branch defects can also be
eliminated in this solution region. After eliminating the defects, we
obtained the feasible solution region as shown in Fig. 6b. In the feasible
solution region, if we want crank-rocker mechanisms, αc of
x-axis must be taken values in range 1 (0∘, 19∘), range
2 (185∘, 228∘) and range 3 (262∘,
360∘), as shown in Fig. 7.
We select αc values from range 1, range 2 and range 3 (see
Fig. 6b) by 5∘ of the step size. Then 247 real exact solutions for
RCCC linkages are obtained by Bertini software which meet the design demands.
Displaying these solutions on the feasible solution region, as shown in
Fig. 7. If we take a smaller step size , the more RCCC linkage solutions can
be obtained. So infinite RCCC linkages can be obtained if we take
infinitesimal step size.
An RCCC linkage can be synthesized by taking a point on the solution region
as shown in Fig. 7. For showing the specific location of RCCC linkage
solutions, let x-component of m1 as x-axis; Let x-component of
m1∗ as y-axis, the solution region shown in Fig. 7 can be
expressed as in Fig. 8. Because the ranges of them are constrained in
[-100,100], so that we have only 246 solutions.
Selecting point p1 as a solution of the RCCC linkage, the results of
solutions are listed in Table 2, the four configurations of the linkage are
shown in Fig. 9. Linkages in Figs. 9 and 10 are automatically generated by
our programmed software using visual language VC++ 6.0 and drawn by OpenGL
software.
Selecting another point p2 as a solution of the RCCC linkage, the
results of solutions are listed in Table 3, the four configurations of the
linkage are shown in Fig. 10.
Discussions
The methodology proposed in this paper synthesizes RCCC linkages by the
solution region theory and Bertini software based on the homotopy method.
Compared with optimization method and iterative method proposed in other
papers, the advantage of the methodology is that more solutions can be
obtained. The optimization method and iterative method usually obtain
approximate solution, while this method can obtain the exact solutions.
Conclusions
We synthesize RCCC linkages based on the solution region methodology for four
specified poses. Before the synthesis of RCCC linkage, we build the
spherical 4R linkage solution region to classify the mechanism types and
eliminate the solutions with defects to reduce the calculations.
The key contributions of this paper are:
All solutions of RCCC linkages for four specific poses are obtained. For
four poses problem, only one RCCC linkage can be synthesized by the methods
published before.
Systematic methodology is proposed in this paper, that is solution
region methodology. Two illustrative examples prove that the method is
effective.
The two synthesized RCCC linkages from the example section have reasonable
size and are crank rocker mechanism which would simplify the control of drive
system. The method not only synthesize more accurate solutions for RCCC
linkage, but also provide more choices for designers. Using solution region
map, the designer can filter the solutions by adding different design
conditions besides the four specified poses. The solution region also can
display some kinematic properties and geometric dimensions of linkages. These
functions are useful for the practical application.
All the data used in this manuscript can be obtained on request from the corresponding author.
JYH proposed the idea and methodology; YC derived the equations and developed the software.
The authors declare that they have no conflict of
interest.
Acknowledgements
This study is supported by the National Natural Science Foundation of China
under grant no. 51775035 and no. 51275034. Thanks to Supercomputing Center of
Chinese Academy of Sciences for its computing service. Edited by: Doina Pisla Reviewed by: two
anonymous referees
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