Modelling a 6 DOF Electromechanical Platform

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Oct 30, 2013 (3 years and 7 months ago)

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Modelling a 6 DOF Electromechanical Platform

R. PEREIRA J.C. QUADRADO
Departamento de Engenharia Electrotécnica e Automação
Instituto Superior de Engenharia de Lisboa

Rua Conselheiro Emídio Navarro, ISEL, 1950-072 LISBOA

PORTUGAL


Abstract: - In this paper the kinematics and dynamic model of a six-degree-of-freedom, 6-DOF, Platform using
electromechanical actuators is presented. A 6 DOF platform in development displays the advantages of using
electromechanical actuators. The kinematics and dynamic of an universal, prismatic and spherical joints set,
known as UPS set is modelled. Kinematics and dynamics of the platform are also included. Several details of
this platform are depicted.

Key-Words: - Electromechanical Platform, 6-UPS, Kinematics modelling, Dynamic modelling, Non-linear
system

1 Introduction
Nowadays the technology used in 6 DOF platforms
has fundamentally two different topologies. One uses
hydraulic actuators and the other uses electric
machines actuators, with mechanical conversion from
rotating to linear displacement.
By analyzing these two different topologies it is
possible to conclude that hydraulic actuators solution
has disadvantages when compared with electrical
machines. The hydraulic actuators are expensive and
require large occupation space.[1] On the other hand,
electric machines with mechanical displacement
adaptation have low operation and maintenance costs
and can lead to the assemblage of a portable system.
The 6 DOF platform in development is a
mechanism with two bodies connected together by
six extendable actuators, known as prismatic joints,
connected through an universal joint at the lower end
and a spherical joint at the upper end. This structure
is referred as 6-UPS [2] (universal-prismatic-
spherical) platform (Fig.1).

Fig.1 – 6-DOF platform in development

Fig.1 represents the actuators placement on the
platform in development.
Different 6 DOF platform architectures depending
on the actuators disposition between two different
bodies, are possible. In [1], the Stewart platform is
described (Fig.2).

Fig.2 - General Stewart platform

The changed Stewart platform (Fig.3) and crossed
leg Stewart platform (Fig.4), are other possible
architectures.

Fig.3 - Changed Stewart platform


Fig.4 - Crossed leg Stewart platform

The changed Stewart platform and the general
Stewart platform are similar. The difference is that in
the first one, the actuators are asymmetrical fixed on
both bodies (Fig.3). Using actuators with the same
length, this architecture provide a large workspace in
comparison with the latest (Fig.2). Crossed leg
architecture provides more rigidity to the structure
then architectures shown in Fig.2 and Fig.3.
The main purpose this work is to obtain the
kinematics and dynamic model of a 6 DOF platform
with an UPS set, to design the closed-loop of the 6-
DOF actuators system.


2 Kinematics and dynamics of the
platform
In this section the kinematics and the dynamics of an
UPS set and the kinematics and dynamics of the
platform are presented.

Fig.5 – UPS set
Fig.5 shows the three joints: an universal joint at
the lower part of UPS set, a prismatic joint given by
the rotor of the electric machine, and a spherical joint
at the upper part of UPS set, where the rotation
movement of this spherical joint is given by the
rotational movement of the rotor.


2.1 Kinematics of an UPS set
The position, velocity and inertia transformation of
an UPS set is given in [2] and is briefly summarized
below.
To an arbitrary UPS set, quantities like, b
i
, q
i
, p
i
,
etc., are simply written as b, q, p, etc., respectively
without subscription i for the sake of written
simplicity.
Starting with position and velocity analysis, the
UPS set vector (S) is given by (1).
S q t b= + −

(1)
The UPS set length (L) is obtained from (1)
according to (2).
L S=

(2)
Unit vector (s) along the leg is given by (3).
S
s
L
=

(3)
The transformation matrix from the universal joint
frame (base-connection-frame) to the base frame can
be written as (4).
ˆ ˆ
ˆ
[ ]T x y z
=

(4)
Where
ˆ ˆ
ˆ
, and
x
y z
are given by (5), (6) and (7),
respectively.
ˆ
x
s=

(5)
(
)
ˆ
k s
y
k s
×
=
×

(6)
ˆ ˆ
ˆ
z x y
=
×

(7)
The COG of the lower and upper part of UPS set,
expressed in the UPS frame is obtain accordinglly to
(8) and (9).
0d d
r Tr
=

(8)
( )
0u u
r T v r= +

(9)
In (9), v is given by the expression (10).
[ 0 0]
T
v L=

(10)
The moments of inertia of lower and upper part of
the UPS set can be written as (11) and (12),
respectively.
0
T
d d
I
TI T=

(11)
2
0
(0,1,1)
T
d d u
I
T I m L diag T


= +


(12)
The velocity of a generic platform point can be
obtained by the expression (13).
S q t
ω
=
× +



(13)
The sliding velocity between the two parts of the
leg, is given by (14).
.L s S
=



(14)
The angular velocity of the UPS set can be written
as given in (15).
S
W s
L
= ×


(15)
The acceleration analysis comprises the linear and
the angular acceleration analysis. The acceleration of
the platform-connection-point of the UPS set is given
by (16) [2].
( )
S t q qα ω ω
=
+ × + × ×



(16)
Equation (16) can also be written in terms of
sliding acceleration at the prismatic joint (17).
( )
2
S Ls W W S W Ls A S= + × × + × + ×

 

(17)
In (17), A is the angular acceleration of the UPS set
and is given by (18).
( )
( )
1 1
2
A
s t q s q L
L L
α
ω ω ω⎡ ⎤
= × + × + × × × −
⎣ ⎦


(18
)


2.2 Dynamics of an UPS set
Considering the rotational equilibrium of the entire
UPS set, the Euler’s equation for the considered UPS
set is obtain (19).
u s
M
s S F C+ × =

(19)
In (19) M
u
is a scalar unknown that is mathematical
eliminated by performing a cross product with s at
both sides of equation (19). Fs represent a joint
reaction given by (24) and C is given by (20).
( )
( ) ( )
'
d d d u u u d d u u
d u d u u
C m r a m r a m r m r g
I
I A W I I W C W f
= × + × − + × +
+ + + × + + +

(20)
In (20), a
d
, a
u
and f, are given by (21), (22) and
(23), respectively.
( )
( )
( )
d
1 1
2 r
d d
d
a s t q r s q
L L
L r
α ω ω
ω ω ω
⎡ ⎤
= × + × × + × × × −


⎣ ⎦

− × + × ×



(21)
( )
( )
( )
( )
{
( )
u
1
.
1
.
2 r 2
d u
u
a s s t q s t q r
L
s s q s q
L
L r L
α α
ω ω ω ω
ω ω ω ω
⎡ ⎤ ⎡ ⎤
= + × × + × × +
⎣ ⎦ ⎣ ⎦
+ × × + × × × −
⎡ ⎤ ⎡ ⎤
⎣ ⎦ ⎣ ⎦

− × + × × +

 
 

(22)
( )
s
f C W
ω
= −

(23)
In (23) f represents the moment of viscous friction
at the spherical joint.
F
s
, after a few mathematical simplifications [2] is
given by (24).
s
F Qt Qq V sF
α
= − + −



(24)
In (24), Q,
q

and V are given by the expressions
(25), (27) and (28), respectively.
(
( )
( )( ) ( )( )
( )
2 2
2
2 2
3
2
2
2.
1
1
..
1
-
-
T
u d d u u
u
T T
u
d d u u u u
T T
d d d u d d
d u
s r m r m r
Q m t ss
L L
m
m r m r E s r r s
L L
m s r s r m s r s r
L
s I I s
⎡ ⎤
+
⎛ ⎞
= + − +
⎢ ⎥
⎜ ⎟
⎝ ⎠
⎣ ⎦

+
+ − + −


×
× + × × −


+
 
(25
)
E
3
is 3×3 identity matrix and
s

is defined by (26).
0
0
0
z y
z x
y x
s s
s s s
s s





= −









(26)

q

is given by (27).
sinq q
=


( )
,
q
α

(27)
And V by (28).
( )
( )
{
( )
}
}
{
( )
{
( )
}
( )
{
( )
{ }
( )
}
( )
[
( )
u
u
1
.
2 r 2..
1 1
2
.
1
2 r
1
2.2
u
u p u
d d d
d u u
u
d u
V m s s s q s q
L
L
r L C L m s g s
s m r s q L r
L L
r m r s s q
s q L r
L
L I I s q
L
ω ω ω ω
ω ω ω ω
ω ω ω
ω ω ω ω
ω ω ω ω ω
ω ω ω


=
× × + × × × −
⎡ ⎤ ⎡ ⎤
⎨ ⎨
⎣ ⎦ ⎣ ⎦




× + × × + + − −



× × × × × − × +⎡ ⎤
⎣ ⎦

+ × × + × × × +
⎡ ⎤
⎣ ⎦

+
× × × − × + × × +
⎡ ⎤
⎣ ⎦

+ + + × × × −
  



( ) ( )
}

d u d d u u s
L
I I m r m r g CW f
ω
ω ω

+

× + − + × + +


(28)

2.3 The platform model
The COG position vector (R) in the base reference
frame can be obtained from the rotation matrix by
(29).
0
R
R
=


(29)
In (29) R
0
is the position vector of platform’s
COG, including payload, in the platform reference
frame [2].
According to [2], the gravity centre acceleration is
given by (30).
( )
a R R t
α ω ω
=
× + × × +


(30)
The inertia moment I, including payload, in the
base reference frame is given by (31).
T
p
I
I= ℜ ℜ

(31)
In (31) I
p
is the inertia moment of the platform in
the platform reference frame.
Considering a external force F
ext
and a external
moment M
ext
acting on the platform, a generic joint
reaction (Fs)
i
(32) and a generic frictional moment f
i

(33) can be calculated.
( )
s i i i
i
F Qt Qq V sFα= − + −



(32)
( )
i s i i
f C W
ω
= −

(33)
The Newton’s equation of the platform can be
written as (34).
( )
6
1
0
ext s
i
i
Ma Mg F F
=
− + +ℜ − =


(34)
After a few mathematical simplifications, (34) can be
written in matrix form (35).
ext
ext
F
t
J HF
M
η
α

⎡ ⎤
⎡ ⎤
+ = +
⎢ ⎥
⎢ ⎥

⎣ ⎦
⎣ ⎦


(35)
In (35) J and η are given by (36) and (37),
respectively.
6
1
plat i
i
J J J
=
= +


(36)
6
1
plat i
i
η
η η
=
= +


(37)
In (37) J
plat
and J
i
can be obtained, respectively, by
(38) and (39).
( )
3
2
3
plat
T
ME MR
J
MR I M R E RR
⎡ ⎤

=
⎢ ⎥
+ −
⎢ ⎥
⎣ ⎦



(38)

i i i
i
i i i i i
Q Qq
J
Qq q Qq

⎡ ⎤
=
⎢ ⎥

⎣ ⎦

  

(39)
In (38),
R

is given by (40)
sinR R= −


(
)
,
R
α

(40)
In (37) η
plat
, η
i
, H and F can be obtained,
respectively, by (41), (42), (43) and (44).

( )
{
}
( )
{ }
.
plat
M R g
I MR R
g
ω ω
η
ω ω ω ω
⎡ ⎤
× × −
=
⎢ ⎥
× + × −
⎢ ⎥
⎣ ⎦

(41)
i
i
i i i
V
q V
f
η


=


× −



(42)
1 2 3 4 5 6
1 1 2 2 3 3 4 4 5 5 6 6
s s s s s s
H
q s q s q s q s q s q s


=


× × × × × ×


(43)

[
]
1 2 ㌴ 3 6
T
F F F F F F F=

(44)


3 Model Application
In order to obtain the kinematics and dynamic model
of the platform in development, the variables are
defined as a first approach by the following
expressions (45-66) for each UPS set [2].
( ) ( )
0.3cos/3 0.3sin/3 0
T
i
b i i
π π
=




(45)

( ) ( )
0.15cos/3 0.15sin/3 0
T
i
p i iπ π=





(46)
[
]
1 ㈳ 2 5 6
k k k k k k =

8.41 0.23 0.95 1.00 0.71 0.95
0.27 0.92 0.29 0.00 0.71 0.29
0.00 0.31 0.09 0.00 0.00 0.09
− −




=







(47)
[
]
0
〮㐰 〮ㄴ 〮ㄸ
T
d
r = −

(48)
[
]
0
㘮6 0⸰.0⸰.
T
u
r = − −

(49)
0
0.010 0.005 0.007
0.005 0.002 0.003
0.007 0.003 0.010
d
I




=







(50)
0
0.005 0.002 0.002
0.002 0.002 0.001
0.002 0.001 0.003
u
I




=







(51)
0.050 0.003 0.004
0.003 0.040 0.003
0.004 0.003 0.100
p
I




=







(52)
[
]
0
〰 0
T
R =

(53)
3
d
m
=
and
1
u
m =

(54)
40
M =

(55)
3
1 10
u
C

= ×
,
2
1 10
p
C

= ×
and
3
2 10
s
C

= ×

(56)
The simulation model initial conditions are given
by (57-60).
[
]
0
0.0 0.0 0.8
T
t m=

(57)
[
]
0
0.0 0.0 0.8/
T
t m s=


(58)
[
]
0
0.0 0.0 1.0
T
radΘ = −

(59)
[
]
0
0.0 0.0 1.0/
T
rad s
ω
= −

(60)


4 Conclusion
In this paper the kinematics and dynamic model of a
6 DOF platform are described. This model can be
applied at any 6 DOF platform varying their own
variables according to the intended system.
In the 3
rd
section, the parameters of the platform in
development are presented. Future work includes
simulation and validation of the obtained model.
Therefore, this work can be used as a base of control
of the 6-UPS simultaneously.


5 Nomenclature
q
i

ℜp
i

Rotation matrix (orientation of platform)
p
i
i-th platform point
t Translation vector (position of platform)
b
i
i-th base point
(r
d0
)
i
Centre of gravity of lower i-th UPS set
(r
u0
)
i
Centre of gravity of upper i-th UPS set
(m
u
)
i
Masses of upper part of i-th UPS set
t


Linear velocity of reference point of platform
t


Linear acceleration of reference point of platform
α Angular acceleration
(m
d
)
i
Masses of lower part of i-th UPS set
C
u
Coefficient of viscous friction in the universal
joint
C
s
Coefficient of viscous friction in the spherical
joint
C
p
Coefficient of viscous friction in the prismatic
joint
F
i
Input force at i-th UPS set
(I
d
)
i

Inertia moment of lower part of i-th UPS set
(I
p
)
i
Inertia moment of upper part of i-th UPS set
g Gravitational acceleration


References:
[1] F. Barata, R. Luís, R. Pereira, Plataforma
Electromecânica de Emulação de um Sistema de
Voo; Projecto Final de Curso, ISEL, 2002. (in
Portuguese).

[2] Bhaskar Dasgupta, T.S. Mruthyunjaya, Closed-
form dynamic equations of the general Stewart
platform through the Newton-euler aproach,
Elsevier Science Lda, Mechanism and Machine
Theory, Volume 33, Issue 7, pages 993-1012,
1998.