# The Displacement Analysis of a Concentric Double Universal Joint

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```					12th IFToMM World Congress, Besançon (France), June18-21, 2007

The Displacement Analysis of a Concentric Double Universal Joint
C.-C. Lin*                                W.-K. Liao
Department of Mechanical and Mechatronic Engineering
National Taiwan Ocean University
Keelung, Taiwan

Abstract— The displacement analysis of a concentric                A comprehensive list of references about constant-
double universal joint is studied. Based on the geometric          velocity joint can be found in the book by Duditza [2].
characteristic of the angle bisector mechanism, the proposed       The constant velocity ratio behaviour of a coupling
method divides the main loop into two artificial loops, and then   between two shafts was first clarified using screw theory
the loop equations are formulated and solved. All the joint
displacements can be expressed in the closed-form functions.
by Hunt [3] in 1973. In that paper he proposed the concept
From the displacement analysis result, it is concluded that the    of the homokinetic plane or bisecting plane, and proved
Thompson CV coupling is a true constant velocity shaft coupling    the validity of the constant-velocity effect when the
with all revolute joint in it.                                     intersection (or the midpoint of the common normal) of
two shaft axes always maintains on the bisecting plane.
Keywords: double universal joint, constant velocity, shaft      The CV couplings such as 7R (double universal joint),
coupling, homokinetic plane, shaft misalignment                    Tracta, Clemens, Myard, and Altmann couplings are
spatial linkages with revolute joints only or combinations
of revolute, prismatic, cylindrical, planar, and spherical
I. Introduction                                                    joints. Some researchers have investigated the kinematics
of the above CV couplings in the past four decades.
It is well understood that a single universal joint can not      Wallace, et al [4] studied the displacement analysis and
maintain a constant velocity ratio between two shafts if           parameter sensitivity analysis of Tracta and Clemens
they are misaligned. A common method to solve the                  couplings. Fischer [5] introduced dual number method to
above problem is to combine two universal joints back-to-          derive the displacement equation for each joint of the
back at two ends of an intermediate link, forming a so-            Tracta coupling. Jun Nango, et al [6] considered the
called double universal joint, and then the variation of the       displacement and static problems of a 7R CV coupling
velocity ratio of each universal joint can be cancelled out.       based on dual number method, and the effects of velocity
Double universal joints have been widely used in the               ratio variation due to manufacturing and assembly errors.
automotive industry for transmitting power between two             Chen [7] studied the kinematics and statics problems of
intersecting or parallel shafts which require a constant           7R, Tracta, and Altmann couplings, and derived the
velocity ratio between them. The main drawback of a                closed-form formulation of the joint displacement and
double universal joint is that the mechanism is relatively         joint torque. Unlike the above mechanisms with one loop
bulky compared to other CV (constant-velocity) couplings           only, the Thompson CV coupling considered here is a
such as Rzeppa, Bendix-Weiss, tripod, etc. The spatial             spherical mechanism with up to twelve links and five
constraint is in general a major concern for a designer in         loops. Thus the displacement analysis problem is much
considering which type of mechanisms should be adopted.            more complicated than that of a single-loop mechanism.
For this reason, the Rzeppa or tripod couplings are most           Wampler [8] presented a modification of Sylvester’s
widely adopted between the shafts in an automotive front-          dialytic elimination process to solve the displacement
wheel drive system. However, Rzeppa or tripod couplings            problem of spherical mechanisms with up to three loops,
also suffer the disadvantage in that considerable amount           and such formulations are without any extraneous roots.
of friction loss or heat dissipation will be generated from        In this paper we shall consider the Thompson CV
the roll-slide motion between the balls and the groove. To         coupling (a five-loop spherical mechanism) and perform
overcome the disadvantages, a new type of CV coupling              the displacement analysis of the mechanism.
called Thompson CV coupling was invented recently [1].
The Thompson CV coupling is essentially a concentric               II. Formulization of joint displacements
double universal joint which is a spherical linkage                A. Modelling of the concentric double universal joint
comprises all revolute joints in it. The mechanism is
relatively compact in size and generates less friction loss          Figure 1(a) shows the computer solid model of a
and wear during operation.                                         Thompson CV coupling. The spherical mechanism has
_________                                                          twelve links interconnected by sixteen revolute joints,
* E-mail: cclin@euler.me.ntou.edu.tw                               resulting in five independent loops and a mobility of one.
12th IFToMM World Congress, Besançon (France), June18-21, 2007

We designate link 2 as the input and link 12 as the output.           (intersection of all joint axes) of the mechanism for each
Both input and output shafts pivot about their joint axes             joint. We first attach a global coordinate system x1y1z1 to

(a)

(a)

(b)
Fig. 1. (a)The solid model of a Thompson CV coupling
(b)The angle bisector mechanism (double spherical parallelogram)

misalignment between the two joint axes is (π − α1). In
Fig. 1(b), we have intentionally hidden the links 10, 3, and
(b)
11 to show the angle bisector mechanism. The angle
bisector mechanism comprises six links (links 6, 4, 9, 7, 5,             Fig. 2. (a)Graph representation (b)Structural representation of the
mechanism (in planar form)
8), forming a double spherical parallelogram. Links 6, 7, 8,
and 9 have equal twist angle, which is also one-half of the           the ground link, where z1-axis is aligned with the joint
twist angle of link 4 or 5. The angle bisector mechanism is           axis of link 2 about link 1, and x1-axis is chosen arbitrarily
pin-supported by links 2, 12, and 3 at points A, B, C,                as long as it is perpendicular to z1-axis. All local
respectively. All joint axes intersect at the stationary              coordinates are then defined subsequently as we traverse
sphere centre O. It can be observed from Fig. 1(b) that the           around each loop. For example, as shown in Fig 2(b),
angle between   OA and OC always equal to the angle                   starting from link 2, a clockwise path is traversed along
R2-R10-R4-R6-R7-R2. The z2-axis is defined along the R2
between OB and OC at any configuration. Therefore                     joint axis, and x2-axis is obtained by the cross product
OC will bisect the angle between z1 (input axis) and z12               z1 × z2 . Then z10-axis and x10-axis are defined in the same
(output axis). By Hunt’s definition in ref. [3], OC is the            manner. The subsequent local coordinate systems in the
loop, x4y4z4, x6y6z6, and x7y7z7 can also be defined
normal of homokinetic plane (bisecting plane). Figures
accordingly. Figure 3 shows only some local coordinate
2(a) and 2(b) show the graph representation and structural
systems due to limited space. Figure 4 shows the layout of
representation of the mechanism, respectively. Links 1, 2,
the input (z1), output (z12), bisecting vector (z4), and the
10, 3, 11, and 12 form a closed-loop kinematic chain
homokinetic plane for the configuration shown in Fig.
(main loop); in which there are two combined universal
1(a). The DH parameters of each link can be defined
joints. To model the multiloop spherical mechanism, a set
of local coordinate system are defined at the sphere centre           according to the convention. For examples, θ2 is the angle
required to rotate the x2-axis into alignment with the x10-
12th IFToMM World Congress, Besançon (France), June18-21, 2007

axis about the positive z2-axis; α10 (or α10,2) is the angle             make the problem solvable, a geometric characteristic of
required to rotate the z2-axis into alignment with the z10-              the double parallelogram is utilized. That is, joint axis z4
axis about the positive x10-axis. Since there are some                   can be considered fixed when the angular misalignment
multiple-jointed links in the CV coupling, more than one                 (π-α1) is kept fixed. As depicted in the previous section
twist angles (αia, αib, ...) can be defined for those links.             about the angle bisector mechanism, the joint axis z4
(along OC ) bisects the angle between z1 and z12 axes. As
shown in Fig. 5, the homokinetic plane bisects the main
loop into two artificial loops L1 and L2. Since z4-axis can
be regarded as being attached to the ground when α1 is
fixed, the twist angle between z1 and z4 axes is also a
constant (α1/2). Hence we can solve the loop equation of
L1 easily. The solution process begins by giving the input
angular displacement θ1. Then from the loop equations of
the four-link chain L1, the rest three angular displacements
(θ2, θ10, θ4g) can be obtained in closed form; where θ2
and θ10 represent the angular displacements at joints R2
and R10, respectively; and θ4g denotes the angular

Fig. 3. The local coordinate system for some joints

Fig. 4 The bisecting axis of the angle between input and output axes   Fig. 5. The loops of the mechanism. The main (bottom) loop is divided
into two virtual loops L1 and L2 by the homokinetic plane.
B. Solution scheme of displacement analysis
Next from loop equations of L3, θ4 can be derived first,
Figure 5 shows that there are five loops inside the
and then θ6 and θ7 can be obtained by back substituting θ4
Thompson CV coupling. The displacement analysis
into the loop equations. Next from loop equations of the
problem will become very complicated and unfeasible to
be formulated if we try to establish the nonlinear loop                  four-link chain L5, θ5 can be derived first, and then θ13 and
equations from the five loops. The most important reason                 θ15 can be obtained by back substituting θ5 into the loop
is that there are too many variables in the main (bottom)                equations. The above process complete the solution of all
loop comprised by six links (1, 2, 10, 3, 11, 12), and the               the angular displacements of the left-hand side loops (L1,
variable elimination process fails in reducing the number                L3, L5). The rest angular displacements of the right-hand
of variables. To simplify the formulation process and                    side loops can be derived in the same manner as the above
process. The solution process starts from the loop L6, and
12th IFToMM World Congress, Besançon (France), June18-21, 2007

then L4, and finally L2. Note that the above solution                  L3 : Ζ7 X 2b Ζ 2 X 10 Ζ10 X 3a Ζ 4 X 4a Ζ6 X 6a = Ι               (7)
scheme is based on the premise of the function of the
angle bisector mechanism.                                                                                ′
L4 Ζ 11 X 12b Ζ 8 X 9a Ζ 9 X 4b Ζ 4 X 3b Ζ 3 X 11 = Ι             (8)
C. Loop equation formulation                                                          ′        ′
L5 Ζ 13 X 6b Ζ 6 X 4c Ζ 5 X 5a Ζ 15 X 7 = Ι                       (9)
In [8] Wampler described a procedure to derive the
displacement equation from each loop equation. Two joint                                      ′
L6 Ζ16 X 5b Ζ 5 X 4d Ζ 9 X 9b Ζ 14 X 8 = Ι                     (10)
variables are eliminated during the process. As shown in
Fig. 5, the input joint axis is z1, and the loop L1 is a four          where Z i' denotes the inverse of the matrix Zi. The
link chain. The loop equation of L1 can be expressed as                occurrence of the inverse matrix is because the paths of
traverse result in opposite sign of a joint angle common to
Ζ 4g X 1a Ζ 1 X 2a Ζ 2 X 10 Ζ 10 X 3a = Ι                      (1)   two neighboring loops.
In the same manner, the rest angular displacements can be
where Xn is the transformation matrix representing the
solved one by one for each loop by back substituting the
side rotation about the xn -axis, which is defined by
previously solved variables to the loop equation (starting
1 0            0                                             from L3, and then L5, L6, L4, L2), finally the output
X n = 0 cα n       − sα n  ,                                 (2)   displacement can be obtained.
                                                               To verify the result, a set of DH parameters shown in
0 sα n
              cα n                                          Table I was assigned to each link of the Thompson CV
coupling. The resulting angular displacements of all joints
and Zn is the transformation matrix representing the joint             in the main loop are shown in Fig. 6 to Fig. 10, where the
angle rotation about the zn -axis, which is defined by                 abscissa is the input joint angle, θ1, which ranges from 0°
cθ n − sθ n 0                                                to 360°, and the shaft misalignment (π – α1) ranges from
Z n =  sθ n cθ n 0
               
(3a)            0° to 30°.
 0      0     1
Link i          twist angle αi           Joint angle θi
               
1a              α1/2                θ4 g
and                                                                          1
1 − t n − 2tn
2
0                                                    1b              α1/2           θ12 (output)
ˆ =                                                                                                                   θ1 (0°~360°)
Zn        2tn       1 − tn 2
0 ,                 (3b)
2
2a              90˚
 0             0     1 + tn 
2                                                   2b              -45˚                θ7
                              
let sin θ n and cos θ n in Eq.(3a) be replaced by the tangent                              3a              90˚                 θ10
3
θ n ).                                                      3b              90˚                - θ4
half-angle function tn (=     tan
2                                                            4a              -30˚                θ4
Let z = [0 0 1]T, the relations z T Z n = z T and Z n z = z                              4b              -30˚                θ9
4
can be obtained. By applying the above relations and Eq.                                   4c              -30˚               - θ6
(3b) to Eq. (1), a scalar equation containing only one
variable t2 can be derived as follows                                                      4d              -30˚                θ5
- θ5
[
z T X 1a Ζ 1 X 2a Ζ 2 X 10 − X 3a (1 + t12 )(1 + t 2 ) z = 0 (4)
ˆ        ˆ            T                   2
]                      5
5a
5b
30˚
30˚                 θ16
After solving t2 from Eq. (4), the joint angle θ2 can be                                   6a              -30˚                θ6
obtained by the equation θ 2 = 2 tan −1 (t2 ) . Pre-multiply Eq.             6
6b              -30˚                θ13
(1) by zT, again we obtain a vector equation containing                      7              7              30˚                 θ15
only one variable θ10 as follows (since θ2 has been solved).
8              8              30˚                 θ14
z T Ζ 4g X 1a Ζ 1 X 2a Ζ 2 X 10 Ζ 10 X 3a = z T I              (5)                       9a              -30˚                θ8
9
Finally, by substituting θ2 and θ10 back into Eq. (1), the                                 9b              -30˚               - θ9
last unsolved variable θ4g in loop L1 can be solved.                        10             10              90˚                 θ2
According to the path of traverse, the loop equations of L2                 11             11              90˚                 θ3
through L6 can be established below.                                                       12a             90˚                 θ11
12
′
L2 Ζ12 X 1b Ζ 4g X 3b Ζ 3 X 11 Ζ 11 X 12c = Ι                (6)                          12b              -45˚                θ11
TABLE I. The DH parameters of the Thompson CV coupling
12th IFToMM World Congress, Besançon (France), June18-21, 2007

Fig. 6. The output joint angle θ12 vs input angle as the misalignment     Fig. 10.The joint angle θ3 vs. input angle as the misalignment changes
changes (four overlapping loci)
III. Result
Figure 6 indicates the angular displacement of the output
shaft is identical to the corresponding input angular
displacement except for a different sign due to the
definition of the local coordinate systems. The four loci
corresponding to four different values of misalignment
(π - α1 = 0°, 10°, 20°, 30°) completely overlap, indicating
the output angular velocity is not affected by the shaft
misalignment. Hence the Thompson CV coupling is
verified theoretically as a true constant velocity shaft
coupling. Some observations from the resulting figures
(Figs. 6~10) are stated below.
Fig. 7. The joint angle θ2 vs. input angle as the misalignment changes    1. The angular displacements of the joints which are
symmetric about the homokinetic plane are identical.
The symmetric angular displacements include: θ1 and
θ12, θ2 and θ11, θ10 and θ3.
2. When the shaft misalignment is zero, all joint angles
are fixed constants except the output joint angleθ12. In
other words, the whole mechanism rotates as a single
rigid body, i.e., no relative motion between inner links.
3. As the shaft misalignment increases, the peak-to-peak
values of the angular displacements also increase.
4. The function of the six-link double parallelogram
mechanisms (angle bisector mechanism) is to provide
the plane of symmetry for the two universal joints. This
is achieved by guaranteeing the equality of the angle
Fig. 8. The joint angle θ11 vs. input angle as the misalignment changes
between     OA and OC and the angle between OB and
OC at any configuration (Fig. 1(b)).
5. The maximum misalignment in this report is 30°,
provided that no interference occurs. However, the
amount of the misalignment can further increase with a
proper design of the link lengths.

Fig. 9. The joint angle θ10 vs. input angle as the misalignment changes
12th IFToMM World Congress, Besançon (France), June18-21, 2007

IV. Conclusion
The displacement analysis problem of a concentric
double universal joint is investigated. Due to the complex
structure of the CV coupling, the solution by using the
formulation of the total five loops is not feasible. Hence
the geometric characteristic of the angle bisector
mechanism is utilized. That is, when the angular
misalignment (π-α1) is kept fixed, the z4-axis of the angle
bisector mechanism can be regarded as a fixed joint axis.
Using the proposed method, the main loop of the coupling
is divided into two artificial loops, and then the loop
equations are formulated and solved. All the joint
displacements can be expressed in the closed-form
functions. From the analysis result, it is concluded that the
Thompson CV coupling is a true constant velocity shaft
coupling with all revolute joint in it. Further study has
been done to optimize the link geometry [9] in order to
allow a larger range of shaft misalignment, and smaller
peak-to-peak values of the joint displacements (under the
same angular misalignment).

References
[1] Thompson CV Couplings website, http://cvcoupling.com
[2] Duditza F. L. Cuplaje mobile homocinetice. Editura Teknica,
Bucharest, 1974.
[3] Hunt K. H. Constant-velocity shaft couplings: a general theory.
Trans. ASME J. Eng. Ind., 95:455-464, 1973.
[4] Wallace D. M., and Freudenstein F. The displacement analysis of the
generalized Tracta coupling. Trans. ASME J. Appl. Mechanics,
37(3):713-719, 1970.
[5] Fischer I. S. Numerical analysis of displacements in a Tracta
coupling. Engineering with Computers, 15:334-344, 1999.
[6] Nango J., and Watanabe K. Static analysis of spatial 7R link
constant-velocity joints extended shaft angle. In Proc. ASME, Des.
Eng. Tech. Conf, Montreal, Canada, 2002.
[7] Chen C. K. The kinematic and static analysis of constant velocity
coupling. Master thesis, Department of mechanical and mechatronic
engineering, National Taiwan Ocean University, 2004. (in Chinese)
[8] Wampler C. W. Displacement analysis of spherical mechanisms
having three or fewer loops. Trans. ASME J. Mech.. Des., 126:93-100,
2004
[9] Liao W. K. The displacement analysis of multi-loop constant
velocity shaft couplings. Master thesis, Department of mechanical and
mechatronic engineering, National Taiwan Ocean University, 2006. (in
Chinese)

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