A Finite and Instantaneous Screw Based Approach for Topology Design and Kinematic Analysis of 5-Axis Parallel Kinematic Machines
Sun et al. Chin. J. Mech. Eng.
A Finite and Instantaneous Screw Based Approach for Topology Design and Kinematic Analysis of 5-Axis Parallel Kinematic Machines
Tao Sun 0
Shuo‑Fei Yang 0
Tian Huang 0 2
Jian S. Dai 1
0 Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University , Tianjin 300350 , China
1 Centre for Robotics Research, School of Natural Sciences and Mathematics, King's College London, University of London , London WC2R 2LS , UK
2 School of Engineer‐ ing, The University of Warwick , Coventry CV4 7AL , UK
Unifying the models for topology design and kinematic analysis has long been a desire for the research of parallel kinematic machines (PKMs). This requires that analytical description, formulation and operation for both finite and instantaneous motions are performed by the same mathematical tool. Based upon finite and instantaneous screw theory, a unified and systematic approach for topology design and kinematic analysis of PKMs is proposed in this paper. Using the derivative mapping between finite and instantaneous screws built in the authors' previous work, the finite and instantaneous motions of PKMs are analytically described by the simple and non‑ redundant screws in quasi‑ vector and vector forms. And topological and parametric models of PKMs are algebraically formulated and related. These related topological and parametric models are ready to do type synthesis and kinematic analysis of PKMs under the unified framework of screw theory. In order to show the validity of the proposed approach, a kind of two‑ translational and three‑ rotational (2T3R) 5‑ axis PKMs is taken as example. Numerous new structures of the 2T3R PKMs are synthesized as the results of topology design, and their Jacobian matrix is obtained easily for parameter optimization and performance evaluation. Some of the synthesized PKMs have outstanding capabilities in terms of large workspaces and flexible orientations, and have great potential for industrial applications of machining and manufacture. Among them, METROM PKM is a typical example which has attracted a lot of attention from global companies and already been developed as commercial products. The approach is a general and unified approach that can be used in the innovative design of different kinds of PKMs.
Innovative design; Parallel kinematic machines; Screw theory; Finite screw; Instantaneous screw
CNC and robot based equipment are important parts
to push “Made in China 2025” plan [1, 2]. In
comparison with traditional CNCs and articulated robots,
parallel kinematic machines (PKMs) are demonstrated by
many researchers to have advantages of high stiffness,
good accuracy, excellent dynamics and reconfigurability
through deeply investigating their topology structures
[3–5], stiffness characteristics , constraint properties
 and kinematic performances [8–10]. They are suitable
to be applied in machining and repairing of large scale
component with complex surface. The innovative design
of PKMs is always a hot topic and draws great attention
from both academia and industry [11, 12].
The innovative design of PKMs usually consists of
topology design and kinematic analysis [13–15]. It has
long been a desire to unify these two parts under a
framework of the same mathematical tool. This mathematical
tool should be able to realize analytical description,
formulation and operation of both finite and
instantaneous motions, and then relate topological and parametric
models algebraically. Till now, there are three available
mathematical tools at hand, i.e., matrix group, dual
quaternion and screw theory.
Matrix group was firstly given out by Lie and later
utilized by Klein to describe rigid body motion. It was
Hervé et al. [16, 17] who gave an approach to formulate
topological models and carried out type synthesis of PKMs
through describing finite motions of PKMs by subgroups
of the matrix representation of the special Euclidean group
) and their composite manifolds. Using this approach,
many PKMs having different motions were synthesized by
Li and Hervé [18, 19]. Describing instantaneous motions
by Lie algebra of SE(
), i.e., se(
), Brockett  applied the
exponential mapping between SE(
) and se(
) to relating
topological and parametric models of open-loop
mechanisms. His work was extended to deal with closed-loop
and other kinds of mechanisms. However, two barriers are
encountered when using matrix groups for finite motion
composition in formulating topological models. One
barrier arises from that matrix group cannot directly reflect
the Chasles’ axis as well as the angular and/or linear
displacement about and/or along that axis. Thus, the
description of finite motion by matrix group is complicated. The
other barrier comes from the incompetent to algebraically
compute the finite motion composition result of matrix
groups by using the Baker-Campbell-Hausdorff formula.
Topological models of many PKMs cannot be simply
written as the group products of a few Lie subgroups of SE(
Hence, type synthesis of these PKMs cannot be precisely
carried out although parametric models for kinematic
analysis can be directly obtained by se(
Dual quaternion can be traced back to the early work of
Euler, Rodrigues and Hamilton. Perez and McCarthy 
are probably the first to use it in the finite and instantaneous
motion analysis of serial kinematic chains. Unit dual
quaternions and unit pure dual quaternions are used by them
to respectively describe finite and instantaneous motions,
because the algebraic structure of the former is a double
cover of SE(
) whose Lie algebra in turn constitutes the
latter. Using group theory, Selig  and Dai  investigated
algebraic properties of the exponential and Cayley mappings
between these two kinds of dual quaternions, resulting in
a clear relationship between the finite and instantaneous
models. It should be noted that even though both finite and
instantaneous motions can be described by dual
quaternions, quaternion representation is not the simplest form. The
redundancy may cause complexity in analytical operations
of finite motions. Additionally, the Rodrigues formula with
dual angles is not the simplest form of the
Baker-CampbellHausdorff formula in composition of finite motions.
Screw theory was firstly proposed by Ball and has been
developed to be a powerful tool in analysis and
mechanical design of PKMs. Instantaneous screw has been proved
to be the simplest and most effective form to describe
instantaneous motion and widely used in formulating
parametric models to conduct velocity and force [24–26],
precision [27, 28] and stiffness [29–32] analysis. In the
authors’ previous work [33–36], finite screw is proved
to be the concise and non-redundant form to describe
finite motion and can be analytically composited by the
screw triangle product . Meanwhile, the algebraic
structures of these two kinds of screws were revealed
and the derivative mapping between them was built by
the authors . All these achievements show that finite
and instantaneous screw theory has the potential to unify
topology design and kinematic analysis into a general and
consistent process by doing type synthesis and kinematic
analysis under this concise mathematical tool, which can
overcome the shortcomings of the above matrix group
and dual quaternion based approaches.
Mainly drawing on finite and instantaneous screw
theory, this paper proposes a unified and systematic
approach for topology design and kinematic analysis of
PKMs. A kind of two-translational and three-rotational
(2T3R) PKMs is taken as example to show the validity
of the proposed approach. These PKMs generate
twoDoF translations in a fixed plane followed by three-DoF
rotations about a fixed point. Some of them have great
potential for industrial applications in 5-axis machining
and manufacture because of their outstanding
capabilities to realize large workspaces and flexible orientations.
METROM PKM is a typical one which has attracted a
lot of attention from global companies and already been
developed as commercial products [37, 38]. The approach
is a general and unified approach that can be used in the
innovative design of different kinds of PKMs.
The paper is organized as follows. Having a brief review
of the state-of-the-art of the existing approaches based
upon different mathematical tools to uniformly describe
finite and instantaneous motions in Section 1, Section 2
presents the theoretical foundations of finite and
instantaneous screw theory. The topological models of PKMs are
formulated by describing the PKMs, their limbs and joints
by finite screws in Section 3, and type synthesis of the
2T3R PKMs is done to show the usages and advantages of
the formulated model. In Section 4, the parametric
models of PKMs are directly obtained through differentiating
the topological models, one typical structure of the 2T3R
PKMs, i.e., the METROM PKM, is selected to show the
detailed procedures. The conclusions of this paper are
drawn in Section 5 (Additional file 1).
2 Screw Theory: Finite and Instantaneous Screws
In this section, we firstly introduce the basic concepts
and properties of finite and instantaneous screws, which
lays the theoretical foundations of type synthesis and
kinematic analysis for innovative design of PKMs.
According to the Chasles’s theorem and Mozzi’s
theorem, both finite motion and instantaneous motion of a
rigid body can be regarded as a rotation about an axis
followed by a translation long that axis, as shown in
Figures 1 and 2.
Figure 1 Finite motion of a rigid body
Figure 2 Instantaneous motion of a rigid body
For finite motion, the axis is referred as Chasles’s axis.
A finite motion is a pose (including orientation and
position) transformation of a rigid body from its initial pose
to arbitrary pose about and along Chasles’s axis. It can be
expressed by a finite screw in quasi-vector form  in
the simplest and non-redundant manner as
Sf = 2 tan + t s , (
where sf and rf denote the unit vector and position vector
of the Chasles’ axis, θ and t are the rotational angle about
and translational distance along that axis with respect to
the initial pose.
For instantaneous motion, the axis is referred as
Mozzi’s axis. An instantaneous motion is a rigid body velocity
measured at a given pose, which is constituted by
angular velocity about and linear velocity along Mozzi’s axis.
For simplicity, it is usually expressed by an instantaneous
screw (twist) in vector form [39–41] as
St = ω
+ v s
where st and rt denote unit vector and position vector of
the Mozzi’s axis, ω and v are angular velocity about and
linear velocity along that axis.
As is well known, successive finite screws of a rigid
body from its initial pose to final pose via several
intermediate poses should be composited in nonlinear
manner. The composition of any two finite screws is expressed
Sf ,ab = Sf ,a △ Sf ,b =
Sf ,a + Sf ,b +
θ θ 0
− tan 2a tan 2b tb sf ,a
1 − tan
a tan θ2b sfT,asf ,b ,
Sf ,b × Sf ,a
+ ta sf ,b
Sf ,a = 2 tan a
Sf ,b = 2 tan b
rf ,a × sf ,a
rf ,b × sf ,b
+ ta sf ,a
+ tb sf ,b
Sf,a and Sf,b are two arbitrary successive finite screws
generated by the same rigid body, the symbol ""∆ is referred
to as screw triangle product and is proven by the authors
of this paper in Ref. .
Using the screw triangle product in Eq. (
resultant finite screw of the rigid body from its initial pose to
final pose can be obtained through computing screw
triangle products of all the successive finite screws it
generates during its continuous finite motion.
Unlike the nonlinear composition of finite screws,
instantaneous screws are composited in linear way.
Suppose a rigid body generates two velocities at a given pose.
Each is expressed by an instantaneous screw as
As proved in our previous work , arbitrary finite
screw that a rigid body generates from its initial pose
can be written in the form shown in Eq. (
its initial pose as the given pose, arbitrary instantaneous
screw the rigid body generates has the form in Eq. (
the initial pose (θ = 0 and t = 0) where the Chasles’ axis
is coincident with the Mozzi’s axis at the instant, there
St,ab = St,a + St,b.
The resultant instantaneous screw can be obtained by
adding them together
exists differential mapping between finite and
instantaneous screws. This means the derivative of finite screw is
derived to be instantaneous screw 
+ v s
This property leads to the algebraic structures of finite
and instantaneous screws. The entire set of finite screws
forms a Lie group under screw triangle product, while
the entire set of instantaneous screws is the
corresponding Lie algebra under screw cross product. The
underlying relationship between these two kinds of screws is
Using this relationship, the topology design and
kinematic analysis of PKMs can be integrated into the unified
framework of screw theory. This is because:
) The topological model of a PKM can be formulated
by describing finite motions of the PKM, its limbs
and joints utilizing finite screws.
) The parametric model of the PKM can be directly
obtained through differentiating its topological
model at given pose.
) Type synthesis and kinematic analysis for
innovative design of PKMs can be easily carried out using
these two models under the simple and consistent
In the following two sections, 5-axis PKMs
having 2T3R motion will be taken as example to show the
detailed procedures of this finite and instantaneous screw
based approach for topology design and kinematic
analysis. Firstly, type synthesis of this kind of PKMs will be
done utilizing finite screws, which will result in
numerous new topology structures with potential industrial
applications. Then, the kinematic analysis of a typical
structure, i.e., the METROM PKM, will be conducted to
show how to directly obtain the parametric model based
upon instantaneous screws through differentiating the
corresponding topological model. The Jacobian matrix
and constraint force will be formulated, which is ready
for parameter optimization and performance evaluation.
3 Type Innovative Design Based upon Finite
3.1 Topological Model of a PKM
Suppose a PKM is composed of l open-loop limbs, as
shown in Figure 3. Each limb consists ni (i = 1, 2, ···, l)
one-DoF joints (revolute joints (R) and prismatic joints
The kth joint
in the ith limb
The ith limb
(P)). Because the finite motions of the PKM’s moving
platform can be obtained as the intersection of those of its
limbs, and the finite motions of each limb are the
composition of those of all joints in it, following analytical
equations can be formulated through describing the finite
motions of the PKM, its limbs and joints as finite screws.
Sf ,PKM = Sf ,1 ∩ Sf ,2 ∩ · · · ∩ Sf ,l,
Sf ,i = Sf ,i,ni △ Sf ,i,ni−1 △ · · · △ Sf ,i,1, i = 1, 2, · · · , l,
k = 1, 2, · · · , ni.
2 tan θ2i,k
Sf ,i,k =
ri,k × si,k
, R joint,
Where Sf,PKM denotes the finite screw generated by the
PKM, Sf,i is the finite screw generated by its ith limb,
Sf,i,k is the finite screw of the kth joint in the ith limb. The
denotations of si,k, ri,k, θi,k and ti,k can be referred to the
symbols in Eq. (
) contain all the topological
information of the PKM, including:
) The number of limbs and the number of joints in
) The type of each joint, i.e., R joint or P joint;
) The direction and position of each joint, i.e., the
geometrical arrangement of each joint in the limb
which it belongs to;
) The geometrical relationships among different limbs.
Thus, these three equations can exactly serve as
topological model of the PKM, which can be used for
topology design. In what follows, we take 5-Axis PKMs having
2T3R motion as example to show the usage of this finite
screw based topological model in doing type
synthesis, resulting in numerous new topology structures with
potential industrial applications.
3.2 Type Synthesis of 2T3R 5A‑xis PKMs
The main goal of type synthesis is inventing innovative
mechanisms with new topology structures having the
given motion pattern. The expected motion pattern of
the discussed PKMs is 2T3R, i.e., two-DoF translations
in a fixed plane followed by three-DoF rotations about
a fixed point O, which allows the PKMs realize 5-axis
machining. Hence, the finite motions of a PKM with this
2T3R motion can be written using finite screw as
Sf ,PKM = t2 s
△ 2 tan b
△ t1 s
where Sf ,LI and Sf ,LII denote the first and second standard Sf,i.
According to Eqs. (
) and (
), joint types and
arrangements of the limb structures which generate the above
two expressions can be obtained. Thus, Eqs. (
) correspond to two standard limbs, RaRbRcP1P2 and
P3RaRbRcP1P2, where the subscripts denote the directions
of the joints. Based upon these two standard limbs, all
the derivative limbs can be synthesized using the
properties of screw triangle product. It should be noted that we
only concern the 5-DoF limbs here, because the 6-DoF
ones can be easily obtained by random permutation of
the joints in P3RaRbRcP1P2.
A 5-DoF derivative limb of RaRbRcP1P2 should satisfy
the following two conditions:
) The finite screw it generates is equivalent to the
standard one, i.e., Sf ,LI in Eq. (
), and thus denoted
hfaacvteodrsiffinerSen′ft,LsIeaqrueetnhcee,soamrSe′fw,LiIthhatshdoisfefeirnent factor(s) with Sf ,LI.
Firstly, we consider the situation that S′f ,LI and Sf ,LI
have the same five factors. It means that the
corresponding derivative limb structures have the same five joints
with RaRbRcP1P2. Hence, these derivative limb structures
can be obtained by permutation of R R R P P
a b c 1 2 while
unchanging the finite screw it generated. Because the
three R joints constituted a spherical joint (S), their
direction can be arbitrarily chosen. Thus, we can suppose that
proved that arbitrarily adjusting the sequence among Rc,
P1 and P2 will always result in the derivative limb
structures that satisfy the two conditions.
For example, R R P P R
a b 1 2 c can be obtained through
changing the orde′r of Rc in RaRbRcP1P2. Based upon
) and (
), S f ,LI generated by RaRbP1P2Rc can be
S′f ,LI = 2 tan θc
△ 2 tan b
rO × sc
According to Eq. (
), the feasible limb structures for the
2T3R PKMs should generate finite screws that contain
Sf,PKM in Eq. (
Through adding none or one translational factor into
Sf,PKM, finite screws of the feasible limb structures should
have either of the following two expressions
Sf ,PKM ⊆ Sf ,i
Sf ,i = Sf ,PKM,
Sf ,i = Sf ,PKM △ t3 s
which correspond to the 5-DoF and 6-DoF limb
Substituting Eq. (
) into Eqs. (
) and (
) leads to the
two standard Sf ,i
Sf ,LI = t2 s
△ 2 tan b
△ t1 s
) has the following equivalent expression
for sc × (s1 × s2) = 0,
Using the properties of screw triangle product, Eq. (
can be rewritten as
r′c = rO − (t1s1 + t2s2) × sc + t1s1 + t2s2
2 tan θ2c
Following the similar way to derive Eq. (
), Eq. (
) can be rewritten as
Sf ,LI = 2 tan θ2c rO × sc + t12st1a+nt2θs2csc2 − (t1s1+t22s2)×sc
△ 2 tan θb
= 2 tan θc
△ 2 tan θa
Because (t1s1 + t2s2) × sc and t1s1 + t2s2 are two
arbitrary orthogonal vectors that are perpendicular to sc,
both rc and rc′ denote arbitrary vectors perpendicular to
sc. Thus, Eq. (
) is equivalent to Eq. (
In this manner, two derivative limb structures,
RaRbP1P2Rc and RaRbP1RcP2, can be synthesized.
Secondly, we consider the situation that S′f ,LI has
different factor(s) with Sf ,LI. In this situation, each
derivative limb structure has at least one different joint with
RaRbRcP1P2. Supposing that the direction of Rc is
perpendicular to that of P1, i.e., scTs1 = 0, it can be proved that
the generated finite screw will not be changed if we use
one Rc to replace P2, or two Rc to replace P1 and P2.
For example, RaRbRcP1Rc can be obtained from
RanaRdb(R8c)P,S1P′f2,LbIygernepelraacteindgbPy2RwaRithbRRcPc.1RAccicsording to Eqs. (
S′f ,LI = 2 tan θc′ sc
2 rQ × sc
where s˜c is the skew matrix of sc, E3 is a unit matrix of
Because scTs1 = 0, the second and third factors are two
translations perpendicular to sc. Using the similar
derivations from Eq. (
) to Eq. (
), the following equivalent
expression of Eq. (
) can be obtained
S′f ,LI = 2 tan θc + θc′
r′′c ×c sc
△ 2 tan θb
rc For andthe r′sca,me r′′creasodnenotaess diasrcbuistrsaerdy abvoeuc-t
tor which is perpendicular to sc, because
t1 exp (θcs˜c)s1 + (exp (θcs˜c) − E3) rQ − rO × sc and
t1 exp (θcs˜c)s1 + (exp (θcs˜c) − E3) rQ − rO are two
arbitrary orthogonal vectors that are perpendicular to sc. This
means that Eq. (
) is equivalent to Eq. (
In this manner, RaRbRcP1Rc and RaRbRcRcRc are
synthesized as derivative limb structures.
Furthermore, it is easy to see that arbitrarily
adjusting the sequence among two Rc and P1 will not change
the generated finite screw based upon the derivations
in the first situation. Hence, two additional
derivative limb structures are obtained, i.e., RaRbP1RcRc and
From the above analysis, totally seven 5-DoF feasible
limb structures for the 2T3R PKMs are synthesized.
For simplicity, we rewrite the three adjacent R joints,
RaRbRc, as S, and the two adjacent R joints, RaRb, as U
(universal joint). These seven limb structures are listed
in Table 1.
Having these limb structures at hand, we can obtain
any 2T3R PKMs with several 5-DoF and 6-DoF limbs
obeying some specific assembly conditions.
) According to Eq. (
), all the 5-DoF limbs in a 2T3R
PKM should have the same translation plane and the
same rotation center O. Thus, only one 5-DoF limb can
be selected to compose a 2T3R PKM because S and/or U
joints that belong to different limbs cannot be placed at a
common point for the convenience of mechanical design.
) In order to design PKMs with suitable actuations,
we select one 5-DoF limb and four 6-DoF limbs to
compose a 2T3R PKM. In this way, each limb has one
actuation. The four 6-DoF limbs are separated into two groups,
which are placed symmetrically with respect to the
translation plane of the 5-DoF limb. The five limbs are fixed
at an icosahedron shape base in order to minimize risk
of collisions and guarantee rigidity of the entire machine
Using these assembly conditions, many innovative
2T3R PKMs can be synthesized. Here, we only list four
topical PKMs due to space limitations, as shown in
Figure 4. Some of these PKMs have been successfully applied
in machining and manufacture or have great potential
industrial applications because of their outstanding
capabilities to realize large workspaces and flexible
orientations, among which the SPR-4(SPRR) in Figure 4(a) is
known as METROM PKM  and has been developed
as commercial product by German company 
(Additional files 2, 3, 4, 5).
4 Kinematic Analysis Based upon Instantaneous
4.1 Parametric Model of a PKM
For a PKM composed of l limbs as shown in Figure 3, its
topological model has been formulated in Eqs. (
based upon finite screws. As discussed in Section 2, the
parametric model for kinematic analysis of the PKM can
be directly formulated using the differential mapping
between finite and instantaneous screws. According to
), the instantaneous screws generated by the PKM,
its limbs and joints can be obtained through
differentiating the corresponding finite screws of them.
Firstly, the instantaneous screw generated by the kth
joint in the ith limb at its initial pose can be obtained
through differentiating Eq. (
Figure 4 Typical 2T3R PKMs. a SPR‑4(SPRR), b SRR‑4(SPRR), c UPPR‑
4(SPRR), d UPRR‑4(SPRR)
) shows the velocity of each one-DoF joint
in the PKM. It can serve as the parametric models of the
joints by taking ωi,k and vi,k as the parameters, because
the unit and position vectors of the joints at their initial
poses are determinate quantities.
In the similar manner, parametric model of the ith limb
can be obtained by differentiating Eq. (
) and obtaining
the instantaneous screws of the limb at its initial pose
St,i = S˙ f ,i θi,k = 0
ti,k = 0 , k=1,2,··· ,ni
Sf ,i,k θi,k = 0
ti,k = 0 , k=1,2,··· ,ni
Finally, the parametric model of the PKM can be
formulated through differentiating Eq. (
St,PKM = S˙ f ,PKM θi,k = 0 i = 1, 2, · · · , l
ti,k = 0 , k = 1, 2, · · · , ni
= S˙ f ,1 θ1,k = 0
t1,k = 0 , k=1,2,··· ,n1
∩ S˙ f ,2 θ2,k = 0
t2,k = 0 , k=1,2,··· ,n2
∩ · · · ∩ S˙ f ,l θl,k = 0
tl,k = 0 , k=1,2,··· ,nl
= St,1 ∩ St,2 ∩ · · · ∩ St,l .
From Eqs. (
), it can be clearly seen that the
relationships between the parametric models of a PKM,
its limbs and joints obtained in this paper are
coincident with those given by other traditional approaches.
However, unlike the traditional approaches, it is
unnecessary to obtain the instantaneous screw system of a
PKM through solving intersection of the instantaneous
screw systems of its limbs in our approach. Based upon
), the instantaneous screw system of the PKM can
be directly formulated through differentiating the finite
screw given in its topological model. In this way, the
Jacobian matrix of the PKM for velocity, force, precision
and stiffness modeling can be easily carried out using the
obtained instantaneous screw system, which is ready to
θ˙i,k ri,k × si,k
St,i,k = S˙ f ,i,k θtii,,kk == 00 = t˙i,k s0i,k ,
ωi,k ri,k × si,k
, R joint
, R joint
conduct kinematic analysis for parameter optimization
and performance evaluation.
4.2 Kinematic Analysis of METROM PKM
Taking a typical structure of the synthesized innovative
PKMs in Section 3, i.e., SPR-4(SPRR) (METROM) PKM
shown in Figure 4(a), for example, the detailed
procedures of how to directly obtain its parametric model
through differentiating the topological model will be
shown. The instantaneous screws related Jacobian matrix
will then be formulated for kinematic analysis.
According to the derivations in Section 3, the
topological model of the METROM PKM can be formulated as
Sf ,METROM = t2 s2
△ 2 tan θb
+ θ˙c rO s×c sc
+ θ˙b rO s×b sb
= v2 s
+ ωb rO s×b sb
+ θ˙a rO s×a sa
+ v1 s
+ ωc rO s×c sc
+ ωa rO s×a sa
It means that the instantaneous screw generated by
the METROM PKM at its initial pose is the linear
combination of five instantaneous screws. Thus, the Jacobian
matrix of this PKM can be obtained through rewriting
) into matrix form
sa sb sc 0 0
rO × sa rO × sb rO × sc s1 s2 ωv1c .
Based upon this Jacobian matrix, the constraint force
exerted on the moving platform of the PKM can be
found. It is a line vector whose axis passes through point
O with the direction s1 × s2, which restrains the one-DoF
translation along its direction. It is expressed by a screw
Sw,METROM = f
rO × s3
, s3 =
s1 × s2
|s1 × s2|
Having the Jacobian matrix and constraint force of the
METROM PKM at hand, kinematic analysis of it can be
carried out. In this way, parameter optimization and
performance evaluation can be done, which are important
parts in the parameter innovative design of METROM
PKM. Because velocity and force [24–26], precision [27,
28] and stiffness [29–32] analysis of PKMs using Jacobian
matrix and constraint force under instantaneous screws
are widely researched, we do not give the detailed
procedures here due to space limitation.
This paper presents a finite and instantaneous screw
based approach for topology design and kinematic
analysis. A kind of 2T3R 5-axis PKMs is taken as example to
show the validity of the proposed approach. Following
conclusions are drawn.
) The topological models of PKMs are formulated by
describing finite motions of the PKMs, their limbs
and joints by finite screws.
) Using the derivative mapping between finite and
instantaneous screws, the parametric models of
PKMs are proved to be directly obtained by
differentiating the corresponding topological models.
) Using these models, type synthesis and kinematic
analysis of PKMs can be carried out and strongly
related. Type synthesis for topology design and
kinematic analysis for parameter optimization and
performance evaluation of a kind of 2T3R 5-axis
PKMs are done to show the validity of the proposed
Additional file 1. Brief Introduction to this paper.
Additional File 2. Vedio of Tricept IV parallel kinematic machine.
Additional File 3. Prototype of Tricept IV parallel kinematic machine.
Additional file 4. Prototype of PaQuad parallel robot.
Additional file 5. Simulation of PaQuad parallel robot.
TS was in charge of the whole trial; TS and S‑FY wrote the manuscript. All
authors read and approved the final manuscript.
Tao Sun, born in 1983, is currently a professor at Tianjin University, China. He
received his PhD degree from Tianjin University, China, in 2012. His research
interests include type synthesis, kinematic/static/dynamic modeling, analysis
and optimal design of parallel mechanisms under framework of screw theory,
orthopedics medical robot. Tel: +86‑22‑87402015; E‑mail: .
Shuo‑Fei Yang, born in 1988, is appointed as a research associate at The
Hong Kong Polytechnic University, Hong Kong, China. He received his PhD
degree from Tianjin University, China, in 2017. His research interests include
type synthesis and kinematic analysis of parallel mechanisms. E‑mail: yangsf@
Tian Huang, born in 1953, is currently a professor at Tianjin University,
China. He received his PhD degree from Tianjin University, China, in 1990. His
research interests include kinematics and dynamics of machine tools and
robotics. E‑mail: .
Jian S. Dai, born in 1954, is currently a professor at King’s College London,
University of London, UK. He received his PhD degree from University of Salford,
UK, in 1993. His research interests include theoretical and computational kin‑
ematics, reconfigurable mechanisms, dexterous mechanisms and manipula‑
tors, end‑ effectors and multifingered hands. E‑mail: .
The authors declare that they have no competing interests.
Ethics Approval and Consent to Participate
Supported by National Natural Science Foundation of China (Grant No.
51675366), and Tianjin Research Program of Application Foundation and
Advanced Technology (Grant Nos. 16JCYBJC19300, 15JCZDJC38900).
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