This document summarizes the kinematics analysis of a 3-UPU (universal-prismatic-universal) parallel robot. Each of the robot's three legs consists of two universal joints connected by a prismatic joint. The document establishes recursive matrix relations for solving the inverse kinematics problem given the position of the mobile platform. Simulation graphs are generated for the input displacements, velocities, and accelerations. The kinematics analysis determines the nine independent variables that define the robot's configuration based on vector-loop equations relating the joint parameters and platform position.

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Kinematics of the spatial 3-UPU parallel robot
Article in UPB Scientific Bulletin, Series D: Mechanical Engineering · July 2013
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2. U.P.B. Sci. Bull., Series D, Vol. 75, Iss. 3, 2013 ISSN 1454-2358
KINEMATICS OF THE SPATIAL 3-UPU PARALLEL ROBOT
Stefan STAICU1
, Constantin POPA2
Recursive matrix relations for kinematics analysis of a parallel
manipulator, namely the universal-prismatic-universal (3-UPU) robot, are
established in this paper. Knowing the translational motion of the platform, the
inverse kinematics problem is solved based on the connectivity relations. Finally,
some simulation graphs for the input displacements, velocities and accelerations
are obtained.
Keywords: Connectivity relations; kinematics, parallel robot
1. Introduction
Parallel robots are closed-loop structures presenting very good potential in
terms of accuracy, stiffness and ability to manipulate large loads. One of the main
bodies of the mechanism is fixed and is called the base, while the other is
regarded as movable and hence is called the moving platform of the manipulator.
Generally, the number of actuators is typically equal to the number of degrees of
freedom and each leg is controlled at or near the fixed base [1].
Compared with traditional serial manipulators, the following are the potential
advantages of parallel architectures: higher kinematical accuracy, lighter weight
and better structural stiffness, stable capacity and suitable position of actuator’s
arrangement, low manufacturing cost and better payload carrying ability.
Accuracy and precision in the direction of the tasks are essential since the
positioning errors of the tool could end in costly damage [2].
Important efforts have been devoted to the kinematics and dynamic
investigations of parallel robots. Among these, the class of manipulators known as
Stewart-Gough platform, used in flight simulators, focused great attention
(Stewart [3]; Di Gregorio and Parenti Castelli [4]). The prototype of the Delta
parallel robot developed by Clavel [5] at the Federal Polytechnic Institute of
Lausanne and by Tsai and Stamper [6] at the University of Maryland, as well as
the Star parallel manipulator (Hervé and Sparacino [7]), are equipped with three
motors, which train on the mobile platform in a three-degrees-of-freedom general
translational motion. Angeles [8], Wang and Gosselin [9] analysed the kinematics,
dynamics and singularity loci of Agile Wrist spherical robot with three revolute
1
Professor, Department of Mechanics, University POLITEHNICA of Bucharest, ROMANIA
2
Lecturer, Department of Mechanics, University POLITEHNICA of Bucharest, ROMANIA

3. 10 Stefan Staicu, Constantin Popa
actuators. In the previous works of Li and Xu [10], [11], the 3-PRC and 3-PUU
spatial parallel kinematical machines with relatively simple structure were
presented with their kinematics solved in details.
In the present paper, a recursive matrix method, already implemented in the
inverse kinematics of parallel robots, is applied to the inverse analysis of a spatial
3-DOF mechanism. It has been proved that the number of equations and
computational operations reduces significantly by using a set of matrices for
kinematics modelling.
2. Kinematics analysis
The 3-UPU architecture parallel manipulators are already well known in the
mechanism community. The manipulator consists of a fixed base 1
1
1 C
B
A , a
circular mobile platform 5
5
5 C
B
A and three extensible legs with identical
kinematical structure. Each limb connects the fixed base to the moving platform
by two universal (U) joints interconnected through a prismatic (P) joint made up
of a cylinder and a piston. Hydraulic or pneumatic systems can be used to vary the
lengths of the prismatic joints and to control the location of the platform (Fig. 1).
Since each U joint consists of two intersecting revolute (R) joints, each leg is
equivalent to a RRPRR kinematical chain. But, the mechanism can be arranged to
achieve only translational motions with certain conditions satisfied, i.e., in each
kinematical chain the axis of the first revolute joint is parallel to that of the last
one and the two intermediate joint axes are parallel to one another. There are three
active mobile prismatic joints and six passive universal joints. The first leg A is
typically contained within the 0
0 z
Ox vertical plane, whereas the remaining legs
C
B, make the angles 0
0
120
,
120 −
=
= C
B α
α respectively, with the first leg (Fig. 2).
For the purpose of analysis, we assign a fixed Cartesian coordinate system
)
( 0
0
0
0 T
z
y
Ox at the centred point O of the fixed base platform and a mobile frame
G
G
G z
y
Gx on the mobile platform at its centre G. The angle ν between 0
Ox and
G
Gx axes is defined as the twist angle of the robot.
The moving platform is initially located at a central configuration, where the
platform is not translated with respect to the fixed base and the origin O of the
fixed frame is located at an elevation h
OG = above the mass centre G.
To simplify the graphical image of the kinematical scheme of the mechanism,
in what follows we will represent the intermediate reference systems by only two
axes, so as is used in most of robotics papers [1], [2], [8]. It is noted that the
relative rotation with angle 1
, −
k
k
ϕ or the relative translation of the body k
T with the
displacement 1
, −
k
k
λ must always be pointed along the direction of the k
z axis.

4. Kinematics of the spatial 3-UPU parallel robot 11
Fig. 1 Symmetric spatial 3-UPU parallel robot
The first active leg A, for example, consists of the cross of a fixed Hooke joint
linked at the frame A
A
A
z
y
x
A 1
1
1
1 , characterised by absolute angle of rotation A
10
ϕ ,
angular velocity A
A
10
10 ϕ
ω
= and the angular acceleration A
A
10
10 ϕ
ε
= , connected at a
moving cylinder A
A
A
z
y
x
A 2
2
2
2 of length 2
l , which has a relative rotation around
A
z
A 2
2 axis with the angle A
21
ϕ , so that A
A
21
21 ϕ
ω
= , A
A
21
21 ϕ
ε
= . An actuated prismatic
joint is as well as a piston of length 3
l linked to the A
A
A
z
y
x
A 3
3
3
3 frame, having a
relative displacement A
32
λ , velocity A
A
v 32
32 λ
= and acceleration A
A
32
32 λ
γ
= . Finally, a
second universal joint A
A
A
z
y
x
A 4
4
4
4 having the angular velocity A
A
43
43 ϕ
ω
= and the
angular acceleration A
A
43
43 ϕ
ε
= is introduced at the edge of a moving platform,
which can be schematised as a circle of radius r in a relative rotation
around A
z
A 5
5 axis with angular velocity A
A
54
54 ϕ
ω
= and angular acceleration A
A
54
54 ϕ
ε
= .
At the central configuration, we also consider that the three sliders are initially
starting from the same position 2
1 sin
/ l
h
l −
= β and that the angles of orientation of
universal joints are given by
3
2
,
3
2
,
0
π
α
π
α
α −
=
=
= C
B
A
,
6
π
ν = (1)
ν
δ
ν sin
tan
)
cos
( 0 r
r
l =
− , δ
β
ν sin
tan
sin h
r = ,
whereδ and β are two constant angles of rotation around the axes A
z1 and A
z2 ,
respectively.
A1
B1
C1
G
A5
B5
C5
O
x0
y0
z0
G

5. 12 Stefan Staicu, Constantin Popa
Fig. 2 Kinematical scheme of first leg A of parallel mechanism
Starting from the reference originOand pursuing along three independent legs
4
3
2
1
0 A
A
A
A
OA , 4
3
2
1
0 B
B
B
B
OB , 4
3
2
1
0 C
C
C
C
OC , we obtain following transformation
matrices
T
i
a
p
p
a
a
p
p 1
21
21
3
10
10 , θ
θ β
ϕ
α
δ
ϕ
=
= , 2
32 θ
=
p , 2
43
43 θ
β
ϕ
a
p
p = , T
p
p 1
54
54 θ
ϕ
= (2)
20
32
30
10
21
20 , p
p
p
p
p
p =
= , 40
54
50
30
43
40 , p
p
p
p
p
p =
= )
,
,
(
),
,
,
( C
B
A
i
c
b
a
p =
= ,
where we denote the matrices [12]:
)
,
( i
i
z
rot
a α
α = , )
,
( δ
δ z
rot
a = , )
,
( β
β z
rot
a =
)
2
/
,
(
1 π
θ x
rot
= , )
2
/
,
(
2 π
θ y
rot
= , )
,
(
3 π
θ y
rot
= , )
,
( 1
,
1
,
i
k
k
k
k z
rot
p −
− = ϕ
ϕ
. (3)
x0
y0
z0
A2 φ21
A
αA
A1
x3
A
z3
A
A3
φ10
A
β
x2
A
G A4
δ
ν
λ32
A
y1
A
O
z1
A
xG
zG
A5
z2
A
φ54
A
z5
A
x5
A
yG
φ43
A
y4
A
z4
A β

6. Kinematics of the spatial 3-UPU parallel robot 13
The angles A
10
ϕ , A
21
ϕ , for example, characterise the sequence of rotations for the first
universal joint 1
A .
In the inverse geometric problem, the position of the mechanism is completely
given through the coordinate G
G
G
z
y
x 0
0
0 ,
, of the mass centre G. Consider, for
example, that during three seconds the moving platform remains in the same
orientation and the motion of the centre G along a rectilinear trajectory is
expressed in the fixed frame 0
0
0 z
y
Ox through the following analytical functions
t
z
z
h
y
y
x
x
G
G
G
G
G
G
3
cos
1
0
0
0
0
0
0 π
−
=
−
=
= ∗
∗
∗
, (4)
where the values *
0
*
0
*
0 2
,
2
,
2 G
G
G
z
y
x denote the final position of the moving platform.
Nine independent variables A
A
A
32
21
10 ,
, λ
ϕ
ϕ , B
B
B
32
21
10 ,
, λ
ϕ
ϕ , C
C
C
32
21
10 ,
, λ
ϕ
ϕ will be
determined by vector-loop equations
G
k
C
C
k
,
k
T
k
C
k
B
G
B
k
k
T
k
B
k
A
G
A
k
,
k
T
k
A
r
r
r
c
r
r
r
b
r
r
r
a
r 5
5
0
4
1
G
1
0
10
4
1
1,
0
10
4
1
1
0
10
5
G
K
G
G
G
G
G
G
G
G
=
−
+
=
−
+
=
−
+ ∑
∑
∑ =
+
=
+
=
+ , (5)
where
0
,
,
)
(
,
0
, 54
3
3
43
1
32
1
32
21
1
1
0
10
G
G
G
G
G
G
G
G
G
G
=
=
+
=
=
= i
i
i
i
i
T
i
i
r
u
l
r
u
l
r
r
u
a
l
r λ
α
).
,
,
(
,
]
0
)
sin(
)
cos(
[
5
C
B
A
i
r
r
r T
i
i
i
G =
+
+
= ν
α
ν
α
G
(6)
From the vector equations (5) we obtain the inverse geometric solution for the
spatial manipulator:
i
G
i
G
i
i
i
y
x
r
l
l α
α
ν
β
ϕ
δ
ϕ
λ cos
sin
sin
)
cos(
)
sin(
)
( 0
0
21
10
32
3
1 +
−
=
+
+
+
+
0
0
0
21
10
32
3
1 sin
cos
cos
)
cos(
)
sin(
)
( l
y
x
r
l
l i
G
i
G
i
i
i
−
+
+
=
+
+
+
+
− α
α
ν
β
ϕ
δ
ϕ
λ (7)
G
i
i
z
l
l 0
21
32
3
1 )
sin(
)
( =
+
+
+ β
ϕ
λ .
The translation conditions concerning the absolute orientation of the moving
platform are given by the following identities
I
R
p
p T
=
=
50
50
D
, )
,
,
(
),
,
,
( C
B
A
i
c
b
a
p =
= (8)
i
T
T
a
a
a
a
t
p
p α
δ
β
β θ
θ
θ
θ
θ 3
1
2
2
1
50
50 )
0
( =
=
=
D
,
where I
R = is the diagonal identity matrix. From these conditions we obtain the
relations between the angles of rotation
i
j
i
i
10
54
21
43 , ϕ
ϕ
ϕ
ϕ =
= , )
,
,
( C
B
A
i = . (9)
The motion of the component elements of the leg A , for example, are
characterized by the relative velocities of the joints

7. 14 Stefan Staicu, Constantin Popa
0
, 1
,
3
32
32
G
G
G
G
=
= −
A
k
k
A
A
v
u
v λ (10)
and by the following relative angular velocities
3
1
,
1
,
32 ,
0 u
A
k
k
A
k
k
A G
G
G
G
−
− =
= ϕ
ω
ω , (11)
which are associated to skew-symmetric matrices
)
5
,
4
,
2
,
1
(
~
~
,
0
~
~
3
1
,
1
,
32 =
=
= −
− k
u
A
k
k
A
k
k
A
ϕ
ω
ω . (12)
From the geometrical constraints (5), we obtain the matrix conditions of
connectivity and, finally, the relative velocities A
A
A
v 32
21
10 ,
, ω
ω of the first leg A [13]:
}
{
~
43
32
32
21
3
10
10
A
T
A
T
T
T
j
A
r
a
r
a
u
a
u
G
G
G
+
ω +
+
+ }
{
~
43
32
32
3
20
21
A
T
A
T
T
j
A
r
a
r
u
a
u
G
G
G
ω 1
20
32 u
a
u
v T
T
j
A G
G G
T
j r
u 0
G
G
= ,
)
3
,
2
,
1
( =
j . (13)
If the other two kinematical chains of the robot are pursued, analogous relations
can be easily obtained.
To describe the kinematical state of each link with respect to the fixed frame,
we compute the angular velocity A
k0
ω
G
and the linear velocity A
k
v 0
G
in terms of the
vectors of the preceding body, using a recursive manner:
3
1
,
0
,
1
1
,
0 u
a A
k
k
A
k
k
k
A
k
G
G
G
−
−
− +
= ϕ
ω
ω , 3
1
,
1
,
0
,
1
1
,
0
,
1
1
,
0
~ u
r
a
v
a
v A
k
k
A
k
k
A
k
k
k
A
k
k
k
A
k
G
G
G
G
−
−
−
−
−
− +
+
= λ
ω . (14)
Rearranging, the above nine constraint equations (7) can be written as three
independent relations
2
0
0 )
cos
sin
sin
( i
G
i
G
y
x
r α
α
ν +
− 2
0
0
0 )
sin
cos
cos
( l
y
x
r i
G
i
G
−
+
+
+ α
α
ν =
+ 2
0 )
( G
z
2
32
3
1 )
( i
l
l λ
+
+
= )
,
,
( C
B
A
i = , (15)
concerning the coordinates G
x0 , G
y0 , G
z0 and the displacements A
32
λ , B
32
λ , C
32
λ only. The
derivative with respect to the time of conditions (15) leads to the matrix equation
T
G
G
G
z
y
x
J
J ]
[ 0
0
0
2
10
1
G
=
λ , (16)
where two significant matrices 1
J and 2
J are, respectively,
}
{
1 C
B
A
diag
J δ
δ
δ
= ,
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
C
C
C
B
B
B
A
A
A
J
3
2
1
3
2
1
3
2
1
2
β
β
β
β
β
β
β
β
β
, (17)
with the notations

8. Kinematics of the spatial 3-UPU parallel robot 15
)
,
,
(
32
3
1 C
B
A
i
l
l i
i =
+
+
= λ
δ
i
i
G
i
l
r
x α
ν
α
β cos
)
cos( 0
0
1 −
+
+
= , i
i
G
i
l
r
y α
ν
α
β sin
)
sin( 0
0
2 −
+
+
= , G
i
z0
3 =
β . (18)
Fig. 3 Input displacements i
32
λ of the three sliders Fig. 4 Input velocities i
v32
of the three sliders
The particular configurations of the three kinds of singularities for the closed-
loop kinematical chains can be determined through the analysis of two Jacobian
matrices 1
J and 2
J [14], [15], [16].
The angular accelerations A
A
21
10 , ε
ε and the relative acceleration A
32
γ of leg A are
expressed by new conditions of connectivity [17]:
}
{
~
43
32
32
21
3
10
10
A
T
A
T
T
T
j
A
r
a
r
a
u
a
u
G
G
G
+
ε +
+
+ }
{
~
43
32
32
3
20
21
A
T
A
T
T
j
A
r
a
r
u
a
u
G
G
G
ε 1
20
32 u
a
u T
T
j
A G
G
γ −
= G
T
j r
u 0
G
G
}
{
~
~
43
32
32
21
3
3
10
10
10
A
T
A
T
T
T
j
A
A
r
a
r
a
u
u
a
u
G
G
G
+
− ω
ω −
+
− }
{
~
~
43
32
32
3
3
20
21
21
A
T
A
T
T
j
A
A
r
a
r
u
u
a
u
G
G
G
ω
ω (19)
}
{
~
~
2 43
32
32
3
21
3
10
21
10
A
T
A
T
T
T
j
A
A
r
a
r
u
a
u
a
u
G
G
G
+
− ω
ω −
− 1
21
3
10
32
10
~
2 u
a
u
a
u
v T
T
T
j
A
A G
G
ω 1
3
20
32
21
~
2 u
u
a
u
v T
T
j
A
A G
G
ω ,
)
3
,
2
,
1
( =
j .
Computing the derivatives with respect to the time of equations (14), we
obtain a recursive form of accelerations A
k0
ε
G
and A
k0
γ
G
:
3
1
,
0
,
1
1
,
1
,
3
1
,
0
,
1
1
,
0
~ u
a
a
u
a T
k
k
A
k
k
k
A
k
k
A
k
k
A
k
k
k
A
k
G
G
G
G
−
−
−
−
−
−
− +
+
= ω
ϕ
ϕ
ε
ε , (20)
3
1
,
3
1
,
0
,
1
1
,
1
,
1
,
0
,
1
0
,
1
0
,
1
1
,
0
,
1
1
,
0
~
2
}
~
~
~
{ u
u
a
a
r
a
a A
k
k
T
k
k
A
k
k
k
A
k
k
A
k
k
A
k
A
k
A
k
k
k
A
k
k
k
A
k
G
G
G
G
G
−
−
−
−
−
−
−
−
−
−
−
− +
+
+
+
= λ
ω
λ
ε
ω
ω
γ
γ

9. 16 Stefan Staicu, Constantin Popa
Fig. 5 Input accelerations i
32
γ of the three sliders Fig. 6 Input displacements i
32
λ of the three sliders
Fig. 7 Input velocities i
v32
of the three sliders Fig. 8 Input accelerations i
32
γ of the three sliders
As an application let us consider a 3-UPU parallel manipulator which has the
following architectural characteristics
m
z
m
y
m
x G
G
G
15
.
0
,
05
.
0
,
05
.
0 *
0
*
0
*
0 =
=
=
s
t
m
h
m
l
A
A
m
l
OA
m
r 3
,
8
.
0
,
6
.
0
,
6
.
0
,
2
.
0 3
4
3
0
1 =
Δ
=
=
=
=
=
= .
Using MATLAB software, a computer program was developed to solve the
kinematics of the 3-UPU parallel robot. To develop the algorithm, it is assumed
that the platform starts at rest from a central configuration and moves pursuing
successively rectilinear translations.
Two examples are solved to illustrate the algorithm. For the first example, the
platform moves along the vertical direction 0
z with variable acceleration while all
the other positional parameters are held equal to zero. The time-histories for the
input displacements i
32
λ (Fig. 3), relative velocities i
v32 (Fig. 4) and relative

10. Kinematics of the spatial 3-UPU parallel robot 17
accelerations i
32
γ (Fig. 5) are carried out for a period of 3
=
Δt seconds in terms of
analytical equations (4).
For the case when the platform’s centre G moves along a rectilinear
horizontal trajectory without any rotation of the platform, the graphs are
illustrated in Fig. 6, Fig. 7 and Fig. 8.
3. Conclusions
Some exact relations that give in real-time the position, velocity and
acceleration of each element of the parallel robot have been established in the
present paper. The simulation certifies that one of the major advantages of the
current matrix recursive formulation is the accuracy and a smaller processing time
for the numerical computation.
Choosing the appropriate serial kinematical circuits connecting many moving
platforms, the present method can be easily applied in forward and inverse
mechanics of various types of parallel mechanisms, complex manipulators of
higher degrees of freedom and particularly hybrid structures, with increased
number of components of the mechanisms.
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11. 18 Stefan Staicu, Constantin Popa
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