The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015
DOI Number: 10.6567/IFToMM.14TH.WC.OS13.131
Mass equivalent triads
V. van der Wijk∗
King’s College London, Centre for Robotics Research, London, United Kingdom
Abstract—In this paper it is shown how a general 3-DoF
triad can be designed mass equivalent to a general (1-DoF)
link element. This is useful in the synthesis of shaking force
balanced and statically balanced mechanisms, for instance
to add or remove a number of DoFs of a balanced mechanism maintaining its balance. Also it can be used as a
simple approach for synthesis of complex balanced mechanisms. To obtain the parameters for mass equivalence,
a mass equivalent model with real and virtual equivalent
masses is used. The characteristics of this model are explained and the properties of a mass equivalent triad are
shown. Subsequently the parameters of a mass equivalent
triad are derived with the method of rotations about the
principal points (RAPP) and application examples are illustrated and discussed.
Keywords: mass equivalent modeling, shaking force balance, static
balance, mass equivalent triad, principal vector linkage
I. Introduction
Figure 1a shows a mechanism link with two joints Ai and
Ai+1 and a center of mass (CoM) in Si , which is a point
defined by parameters ei and fi as illustrated. The mass
of such a general element cannot be modeled mass equivalently with common mass modeling methods such as those
in [8] [2]. This is since these methods only use equivalent
masses in the joints, here Ai and Ai+1 , which means that
only a CoM that is located on the line through the joints
can be modeled. A nonzero value of fi in Fig. 1a then is
not possible.
In [3] a new mass equivalent model of the link in Fig. 1a
was proposed and validated, which is shown in Fig. 1b.
Here a real equivalent mass mai is located in Ai , a real
equivalent mass mbi is located in Ai+1 , and a virtual equivalent mass mci is located√
in both Ji1 and Ji2 . Ji1 is located at a distance si1 = e2i + fi2 from Si normal to the
line A√
i Si , as illustrated, and Ji2 is located at a distance
si2 = (li − ei )2 + fi2 from Si normal to the line Ai+1 Si ,
as illustrated. As in Fig. 1a Si is the CoM of the four equivalent masses.
In [3] [4] it was shown how with this model the motion
of the mass of a general closed kinematic chain can be analyzed as an open chain where one of the elements is considered with equivalent masses. With this method it was
possible to synthesize general inherently shaking force balanced closed-chain linkages which are statically balanced
∗ [email protected]
as well. In [5] it was shown that this mass equivalent model
can represent not only the mass of a general single element,
but also the mass of multi-DoF linkages. For all motion
these multi-DoF linkages then are mass equivalent to the
model in Fig. 1b, which was shown and explained by the
synthesis of a mass equivalent 2-DoF dyad linkage. The
aim of this paper is to extend this theory to the synthesis of
mass equivalent 3-DoF triad linkages. This step requires the
use of principal vectors for calculating the linkage parameters, which for the 2-DoF dyad linkage could be omitted by
using a simpler analysis method.
The application of mass equivalent linkages was shown
in a preliminary study presented in [6] and in [5] where
mass equivalent linkages were applied to add DoFs to a
shaking force balanced or statically balanced mechanism,
maintaining its balance. A balanced 3-DoF parallel manipulator was synthesized from a 0-DoF structure. At the end
of this paper application examples of mass equivalent triads
are shown and discussed. For a realistic interpretation, all
illustrations in this paper are drawn to scale.
(a)
(b)
mci
mci Ji1
Ai
si2
si1
mi S
i
fi
ei
li
qi
Ai+1
ei
~
Ai
Ji2
Si
fi
qi
mbi
Ai+1
li
mai
c
i
m
(c)
c
i
m
Ji1
Ji2
si2*
si1*
Si
*
ei
~
Ai
b
mi
Ai+1
fi*
*
li
qi
a
mi
Fig. 1. a) Link element with two joints Ai and Ai+1 and a general CoM
Si ; b) Mass equivalent model of the link element with a real equivalent
b
mass ma
i in Ai , a real equivalent mass mi in Ai+1 , and a virtual equivb
c
alent mass mci in both Ji1 and Ji2 ; c) ma
i , mi , and mi are independent
of the scale of the link.
II. Mass equivalent triad
The conditions for which the model in Fig. 1b is mass
equivalent with the link element in Fig. 1a are mai ei =
mbi (li − ei ) and mci li = (mai + mbi )fi , which define the
CoM of the model in Si , and mai + mbi = mi which is
the total mass of the model equal to the mass of the link.
Since mass mci = mi fi /li is not part of the ’real’ mass of
the link, this mass is referred to as a virtual equivalent mass
while masses mai = mi (1 − ei /li ) and mbi = mi ei /li are
referred to as the real equivalent masses [3].
In Fig. 1c it is shown that when the model is scaled,
the equivalent masses remain unchanged. The values of
the equivalent masses are independent of the size of the
model. They can be described by mai = mi (1 − λ1 ),
mbi = mi λ1 , and mci = mi λ2 with the constants λ1 = ei /li
and λ2 = fi /li .
III. Method of rotations about the principal points
(RAPP)
The design parameters of the triad for mass equivalence
can be calculated with the method of rotations about the
principal points (RAPP) [3][5]. Figure 3 shows the triad in
Fig. 2 with a graphical construction of parallelograms. This
construction is known as a (3-DoF) principal vector linkage with which the motion of masses mi1 , mi2 , and mi3 is
traced with respect to their common CoM in Si [3]. Of this
principal vector linkage links Ai B1 , B1 B2 , and B2 Ai+1
are the principal elements and P1 , P2 , and P3 are the principal points in each principal element.
mi2
a23 B2
P2
pb2 a
21
mi2
B1
a1
P1
(a)
fi2
ei2
li2
B1
ei1
fi1
mi1
B2
ei3
li3
Si
fi3
mi3
li
Si m
i2
Ai
(b)
mi1
*
li
ei1
Ai
B1
ei2
B2
fi2
B2
Fig. 2. Triad that is mass equivalent with the link element in Fig. 1a in
two poses (drawn to scale with parameters λ1 = 0.5991, λ2 = 0.2658,
li2 /li1 = 1.5540, li3 /li1 = 1.2423, mi2 /mi1 = 0.8947, and
mi3 /mi1 = 1.2632).
While the length of the rigid link in Fig. 1a is fixed, a
released length li∗ as in Fig. 1c can be obtained for instance
as the distance between the extremities of a triad. Figure 2
shows a triad with link lengths li1 , li2 , and li3 where li∗ is
the distance between joints A1 and Ai+1 . Since the link
masses mi1 , mi2 , and mi3 are located such that their common CoM is exactly in Si of the model in Fig. 1c for any
of its poses and mi1 + mi2 + mi3 = mi , this triad is mass
equivalent to the link in Fig. 1a. Practically this could mean
that in a balanced linkage a general single link element can
be substituted with a mass equivalent triad without affecting the balance. Alternatively it could mean that the mass
motion of the triad can be represented by the mass motion
of a single link element.
P2
c2
b21
a3 P3
p3
a23
B2
b23
mi3
Si
Ai+1
li1
li
Ai
mi3
(b)
Si
mi2 p3
mi1
a3
Ai+1
fi1
B1
p1
P1
a1
Ai c1 b1 B1
fi3
ei3
a21
li3
mi1
Ai+1
li1
li2
p1
mi3
(a)
pa2
P3
c3 Ai+1
Ai
P3 Ai+1
*
i
l
P1 a1
a21
P2
a3
a23
B2
B1
b3
Fig. 3. Mass equivalent triad with principal points P1 , P2 , and P3 and its
principal vector linkage which traces the common CoM of mi1 , mi2 , and
mi3 in Si for all motion of the linkage, drawn for two poses.
The principal points are defined within each element with
the principal dimensions a1 , a21 , a23 and a3 , which are the
lengths of the sides of the illustrated parallelograms. Then
P1 is located at an a1 distance from B1 along the line trough
B1 and mi1 and this point is defined relative to line B1 Ai
with parameters b1 and c1 . Similarly P3 is located at an a3
distance from B2 along the line trough B2 and mi3 and this
point is defined relative to line B2 Ai+1 with parameters b3
and c3 . P2 is located at an a21 distance from B1 and an
a23 distance from B2 . This point is defined relative to line
B1 B2 with parameters b21 from B1 and c2 relative to line
B1 B2 , as illustrated.
The parallelograms trace the common CoM in Si for all
motion for the balance conditions [3]
mi1 p1 = (mi2 + mi3 )a1 ,
mi3 p3 = (mi1 + mi2 )a3 ,
mi1 a21 = mi2 pa2
mi3 a23 = mi2 pb2
(1)
Here pa2 and pb2 define a parallelogram through P2 and mi2 ,
aligned with B1 P2 and B2 P2 . Because of similarity, the
mass parameters of each outer principal element are related
as
ei1
fi1
p 1 + a1
=
=
a1
b1
c1
p 3 + a3
ei3
fi3
=
=
a3
b3
c3
(a)
B2
(2)
B1
mci a1
The mass parameters of elements 1 and 3 then can be derived to depend on bi and ci as
ei1
fi1
ei3
fi3
p1
= (
a1
p1
= (
a1
p3
= (
a3
p3
= (
a3
+ 1)b1
+ 1)c1
+ 1)b3
+ 1)c3
mi2 + mi3
=(
mi1
mi2 + mi3
=(
mi1
mi1 + mi2
=(
mi3
mi1 + mi2
=(
mi3
+ 1)b1
+ 1)c1
m
fi2
mi3 li2
mi1 + mi3
+(
+ 1)b21
mi2
mi2
mi1 + mi3
= (
+ 1)c2
mi2
a23 B2
a21
li2
B1
mci
(4)
a1
c
i
m
m
a
i
c
mi
li2
b
mi
Si
a
m i Ai
(c)
B2
Si
B1
.
qi3
xi3
mci
a3 P3
mci
li3
mbi
yi3 Ai+1
m
Ai
Fig. 5. a) Relative motion of DoF 1 with rotational motion about P1 ;
b) Relative motion of DoF 2 with rotational motion about P2 ; c) Relative
motion of DoF 3 with rotational motion about P3 .
mb
Ai+1 i
li
P1
yi2
a
i
li3
Si
li1
a3 P3
xi2
a23 B2
Ai+1
mci
P2
mci
B1
which can be interpreted as that the location of P2 in element 2 is found as the common CoM of mi1 projected in
B1 , mi3 projected in B2 , and mi2 at its actual location.
mci
Ai+1
+ 1)c3
= −
mci
.
qi2
P2
a21
For element 2 the CoM can be described based on P2 as [3]
ei2
li1 xi1
yi1
c
mi
(3)
+ 1)b3
b
mi
Si
P1
mai Ai
c
i
(b)
.
qi1
Ai
Fig. 4. Mass equivalent model of the triad with the projection of the real
b
equivalent mass ma
i in Ai , the real equivalent mass mi in Ai+1 , and the
virtual equivalent mass mci twice about each principal point as illustrated.
To calculate the parameters of the principal points bi and
ci , Fig. 4 shows the mass equivalent model of the triad. The
real equivalent masses mai and mbi are located in joints Ai
and Ai+1 , respectively, and the virtual equivalent mass mci
is located twice about each principal point P1 , P2 , and P3
as illustrated. This projection is explained in [3], and is
for each element of similar composition as the models in
Fig. 1b.
The model in Fig. 4 is analyzed with the linear momentum equations of the three relative motions of the principal
vector linkage, which are the rotational motions about each
principal point P1 , P2 , and P3 individually. An important
property of a virtual equivalent mass is that its mass acts as
a real mass for rotational motion while is has zero mass for
translational motion of its element [3]. This is since the virtual equivalent masses represent solely the rotational mass
motion of an element, while translational mass motion of an
element is represented solely with real equivalent masses.
In Fig. 5 the relative motions are illustrated. Figure 5a
shows the relative motion of DoF 1 where element Ai B1
rotates about P1 and elements B1 B2 and B2 Ai+1 solely
translate. For this motion the virtual equivalent masses in
element Ai B1 act as a real mass while the virtual equivalent
masses in elements B1 B2 and B2 Ai+1 are zero. Figure 5b
shows the relative motion of DoF 2 where element B1 B2
rotates about P2 and elements Ai B1 and B2 Ai+1 solely
translate. For this motion the virtual equivalent masses in
element B1 B2 act as a real mass while the virtual equivalent
masses in elements Ai B1 and B2 Ai+1 are zero. Figure 5c
shows the relative motion of DoF 3 where element B2 Ai+1
rotates about P3 and elements Ai B1 and B1 B2 solely translate. For this motion the virtual equivalent masses in element B2 Ai+1 act as a real mass while the virtual equivalent
masses in elements Ai B1 and B1 B2 are zero.
mci
(b)
m
(a)
B1
mai
b
B1 m i
c
m i a1 b1
c1 li1
mci
a
mi
xi1
xi2
yi2
a23 B2
P2 c2
mbi
a21 b21 l
i2
c
i
(c)
m
a
i
B2
With (1) then p1 , pa2 , pb2 , and p3 are calculated as
c
i
m
i3
Ai
p1 =
mci
a3 P3
b3
l c3
Ai+1
xi3 yi3
mbi
yi1
Fig. 6. Reduced mass models of (a) DoF 1 where P1 is the common
CoM, (b) DoF 2 where P2 is the common CoM, and (c) DoF 3 where P3
is the common CoM. For rotational motion about the principal points the
linear momentum of these models is equal to the linear momentum of the
respective relative motion in Fig. 5.
The linear momentum of the relative motions in Fig. 5
can be represented with the reduced mass models in Fig. 6.
For rotational motion about P1 of the model in Fig. 6a the
linear momentum equations with respect to reference frame
xi1 yi1 are equal to the linear momentum equations of the
motion in Fig. 5a. For rotational motion about P2 of the
reduced mass model in Fig. 6b the linear momentum equations with respect to reference frame xi2 yi2 are equal to the
linear momentum equations of the motion in Fig. 5b. For
rotational motion about P3 of the model in Fig. 6c the linear momentum equations with respect to reference frame
xi3 yi3 are equal to the linear momentum equations of the
motion in Fig. 5c. Therefore these models are also named
Equivalent Linear Momentum Systems (ELMS) [3]. As a
result, the principal points P1 , P2 , and P3 are the common
CoMs of the reduced mass models. Therefore the parameters in each reduced mass model are related by
mai (li1 − b1 ) = mbi b1 ,
mai b21 = mbi (li2 − b21 ),
mai b3 = mbi (li3 − b3 ),
mci li1 = (mai + mbi )c1
mci li2 = (mai + mbi )c2
mci li3 = (mai + mbi )c3
(5)
From these equations b1 , c1 , b21 , c2 , b3 , and c3 are derived
as
b1 =
ma
i li1
ma
+mbi
i
b21 =
b3 =
= (1 − λ1 )li1 ,
mbi li2
ma
+mbi
i
mbi li3
ma
+mbi
i
c1 =
= λ1 li2 ,
c2 =
= λ1 li3 ,
c3 =
mci li1
ma
+mbi
i
mci li2
ma
+mbi
i
mci li3
ma
+mbi
i
Subsequently with (3) and (4) parameters ei1 , fi1 , ei2 , fi2 ,
ei3 , and fi3 are obtained with which the principal dimensions a1 , a21 , a23 , and a3 can be calculated as
√
√
a1 =
b21 + c21 = li1 (1 − λ1 )2 + λ22
√
√
b221 + c22 = li2 λ21 + λ22
a21 =
(7)
√
√
(li2 − b21 )2 + c22 = li2 (1 − λ1 )2 + λ22
a23 =
√
√
b23 + c23 = li3 λ21 + λ22
a3 =
= λ2 li1
= λ2 li2
= λ2 li3
(6)
mi2 + mi3
mi1 + mi2
a 1 , p3 =
a3
mi1
mi3
mi1
mi3
a21 , pb2 =
a23
pa2 =
mi2
mi2
(8)
Substituting these parameters in (3) and (4) with mai +mbi =
mi1 + mi2 + mi3 then leads to
ma li1
,
mi1
mb li2 − mi3 li2
=
,
mi2
mb li3
ei3 =
,
mi3
ei1 =
ei2
mc li1
mi1
mc li2
=
mi2
mc li3
=
mi3
fi1 =
fi2
fi3
(9)
where the mass parameters of the triad are dependent on the
equivalent masses.
IV. Discussion
The method of rotations about the principal points gives
fundamental insight. The principal points in the reduced
mass models in Fig. 6 are related with the so called
’barycenters’ in [7]. Contrary to the barycenters that are
obtained without understanding, the principal points have
physical meaning as being the joints of a principal vector linkage with principal dimensions, as shown in Fig. 3.
Therefore this method may be valuable also for general dynamical analysis. It is interesting to note that the reduced
mass models are of similar composition as the mass equivalent models in Figs. 1b and 1c.
For the calculation of the principal points in the elements
at the extremities, here elements 1 and 3, it is also possible
to use the method of rotations about the principal joints
as shown in [3] [5]. Then the results for P1 and P3 are
obtained quicker and show to be exactly equal to the results
for the principal points of a mass equivalent dyad.
The mass equivalent triad in Fig. 2 can be regarded a
general force balanced 4R four-bar linkage when joints Ai
and Ai+1 are fixed pivots with the base. The common CoM
in Si then is a stationary point in the base. The balance
conditions of this linkage were found in another way in [1].
Figure 7 shows the result of a general force balanced
4R four-bar linkage A0 A1 A2 A3 of which the coupler link
m22
(a)
f22
B1
m21
l2
l3
l1
m4
m23
f22
e22
m21
A1
A0
f1
m1
-e1
S3
m31
m2
f2
l33 B2
l2
e2
l3
m4
m33
A3
A2
A2
S -o2 o1
l4
e4
f4
l1
B2
(a)
f23
l2
l32
m4
A1
A3
l1
e3
m3
f3
Fig. 7. Force balanced 6R six-bar linkage obtained from a general force
balanced 4R four-bar linkage A0 A1 A2 A3 with coupler link A1 A2 substituted with the mass equivalent triad in Fig. 2, illustrated for two poses.
A1 A2 is substituted with the mass equivalent triad in Fig. 2.
Since the distance l2 is no longer fixed, two DoFs are
gained, obtaining a force balanced 6R six-bar linkage of
which the common CoM is in S for all motion. The mass
parameters of the other links, i.e. the CoMs of m1 , m3 ,
and m4 , and the location of the common CoM in S are not
affected by this substitution while force balance is maintained.
In Fig. 8 it is shown how link A2 A3 of a general force
balanced 4R four-bar linkage A0 A1 A2 A3 can be substituted with a mass equivalent triad. For the mass equivalent model of the illustrated triad holds λ1 = 1.5700 and
λ2 = 0.1978. Also here the mass parameters of the other
links are not affected by this substitution while force balance is maintained.
In Figs. 9 and 10 it is shown how the balanced 6R sixbar linkage in Fig. 7, which can also be considered a bal-
A0
m1 f1 -e1
m33
l32
B1
l31
*
e21
f21
S -o2 o1
l4
e4
f4
m1 f1 -e1
e23
B2
l33
B1
f3
m22
S2
l2
l31
A0
m3
A2
A1
e3
l1
A
S -o2 o1 3
l4
e4
f4
m1 f1 -e1
B1
m23
A1
A0
(b)
f23
m2
f2
e2
A2
l21
f21
m32
l22
e23
S2 l23
e22
e21
B2
S -o2 o1
A3
l4
e4
f4
S3
m4
m32
(b)
m31
Fig. 8. Force balanced 6R six-bar linkage obtained from a general force
balanced 4R four-bar linkage A0 A1 A2 A3 with link A2 A3 substituted
with a mass equivalent triad, illustrated for two poses.
anced 2-RRR parallel mechanism, can be synthesized from
a single mass model. The starting point is the mass model
in Fig. 9a, which is equal to the model in Fig. 1a. This
model can be divided in two equivalent mass models as illustrated in Fig. 9b, where the common CoM of both models is in S and their combined mass is equal to the model
in Fig. 9a. By mass equivalent modeling then each of the
two mass models can be substituted with a mass equivalent
triad as shown in Fig. 9c. As a last step the elements at
the extremities can be combined as shown in Fig. 10 where
mp = m13 + m23 . The result is a balanced 2-RRR parallel
mechanism of which the common CoM of the moving links
is in S for all motion.
(a)
B3
(b)
S
A3
A0
B0
S1
S2
S
A0
B0
m13
(c)
m23
B3
A2
m22
A3
B2
m12
B1
A1
S1
Acknowledgement
S2
S
A0
This publication was financially supported by the Niels
Stensen Fellowship.
B0
m11
References
m21
Fig. 9. a) An initial mass model (b) can be divided in two equivalent mass
models (c) which each can be substituted with a mass equivalent triad.
m13
m23
mp
B3
A3
m22
A2
B2
m12
A1
S1
A0
m11
B1
S
V. Conclusion
In this paper it was shown how a general 3-DoF triad can
be designed mass equivalent to a general (1-DoF) link element. For finding the parameters for mass equivalence,
a mass equivalent model with real and virtual equivalent
masses was used. First the characteristics of this model
were explained and subsequently the properties of a mass
equivalent triad were obtained. With the method of rotations about the principal points (RAPP) the parameter values of a mass equivalent triad were derived.
As an application example it was shown how a force balanced 6R five-bar linkage can be obtained from a force balanced 4R four-bar linkage by either substituting the coupler link or by substituting the crank or rocker link with a
mass equivalent triad. It was also shown how a balanced
2-RRR parallel manipulator can be obtained from a single
mass equivalent model by dividing it in two mass equivalent models and by substituting each mass equivalent model
with a mass equivalent triad.
S2
B0
m21
Fig. 10. A balanced 2-RRR parallel mechanism obtained by combining
the links at the extremities of the two mass equivalent triads in Fig. 9c of
which the common CoM is in S for all motion.
[1] Berkof, R.S., Lowen, G.G.: A new method for completely force balancing simple linkages. Engineering for Industry pp. 21–26 (1969)
[2] Chaudhary, H., Saha, S.K.: Balancing of shaking forces and shaking moments for planar mechanisms using the equimomental systems.
Mechanism and Machine Theory 43, 310–334 (2008)
[3] Van der Wijk, V.: Methodology for analysis and synthesis of inherently force and moment-balanced mechanisms - theory and applications (dissertation). University of Twente (free download:
http://dx.doi.org/10.3990/1.9789036536301) (2014)
[4] Van der Wijk, V.: Closed-chain principal vector linkages. In: P. Flores and F. Viadero (eds.), New Trends in Mechanism and Machine
Science 24, 829–837 (2015). Springer.
[5] Van der Wijk, V.: Mass equivalent dyads. In: S. Bai and M. Ceccarrelli (eds.), Recent Advances in Mechanism Design for Robotics
MMS 33, 35–45 (2015). Springer.
[6] Van der Wijk, V., Herder, J.L.: On the addition of degrees of freedom
to force-balanced linkages. Proceedings of the 19th CISM-IFToMM
Symposium on Robot Design, Dynamics, and Control pp. 2012–025
(2012)
[7] Wittenburg, J.: Dynamics of Systems of Rigid bodies. B.G. Teubner
Stuttgart (1977)
[8] Wu, Y., Gosselin, C.M.: On the dynamic balancing of multi-dof parallel mechanisms with multiple legs. Mechanical Design 129, 234–238
(2007)