= J ⋅ M
represents the moment acting on the mobile platform
P( ine)
A
P( ine)
B
E
expressed, this time, in the base frame.
From equation (24) it can be concluded that two matrices playing the roles of the inertia
matrix and the Coriolis and centripetal terms matrix are:
I
⋅ T
(30)
P B
d
(I ⋅T
(31)
P
)
B
dt
It must be emphasized that these matrices do not have the properties of inertia or Coriolis
and centripetal terms matrices and therefore should not, strictly, be named as such.
Nevertheless, throughout the paper the names “inertia matrix” and “Coriolis and centripetal
terms matrix” may be used if there is no risk of misunderstanding.
On the other hand, from equation (26), the inertia matrix and the Coriolis and centripetal
terms matrix, expressed in the Euler angles system, are:
I
= T T ⋅ I
⋅ T
(32)
P
P
E
B
d
V
= T T ⋅
(33)
P
(I ⋅T
P
)
E
B
dt
3.2 Actuators modeling
As the RCID actuators can only move perpendicularly to the base plane, their angular
velocity relative to frame {B} is always zero. So, each actuator can be modelled as a point
mass located at its centre of mass.
The linear momentum of each actuator along direction z B, q , is obtainable from:
i
A
q = m l
⋅
(34)
A
A
i
i
where mA is the mass and l the velocity of actuator i.
i
Simultaneously considering the six actuators results into:
q
⎡ A ⎤
⎡ l ⎤
⎢ 1 ⎥
⎢ 1
q
⎥
l
q
⎢ A 2 ⎥
⎢ 2⎥
=
= m
= m ⋅ l
(35)
A
⎢ # ⎥
A ⎢ # ⎥
A
⎢
⎥
⎢
q
⎥
⎢⎣ A 6 ⎥⎦
⎢⎣ l 6⎥⎦
78
Parallel Manipulators, Towards New Applications
The use of velocity inverse kinematics and transformation T in equation (35) leads to:
B
q = m ⋅ J ⋅ T⋅ x
(36)
A
A
C
P B| E
The inertial component of the actuating forces, τ
, due to actuators translation may be
A( ine)
obtained from the time derivative of equation (36):
τ
= q = m ⋅ J ⋅ x
+J ⋅ x
(37)
A ine
A
A
( B
B
E
P
E
P
)
( )
B| E
B| E
Multiplying equation (37) by T
J the inertial component of the generalized force acting on
C
{P} due to actuators translation, expressed in frame {B}, is obtained as:
P
T
B
T
B
f
= m ⋅ J ⋅ J ⋅ x
+ m ⋅ J ⋅ J ⋅ x
(38)
A( ine)
A
C
E
P
A
C
E
P
B
B| E
B| E
The inertial component of the generalized force acting on {P} due to actuators translation,
expressed using the Euler angles system, is:
P
T P
T
T
B
T
T
B
f
= T ⋅ f
= m ⋅ T ⋅ J ⋅ J ⋅ x
+ m ⋅ T ⋅ J ⋅ J ⋅ x
A( ine)
A( ine)
A
C
E
P
A
C
E
P
B| E
B
B| E
B| E
(39)
T
B
T
B
= m ⋅ J ⋅ J ⋅ x
+ m ⋅ J ⋅ J ⋅ x
A
E
E
P
A
E
E
P
B| E
B| E
The inertia matrix and the Coriolis and centripetal terms matrix, expressed in the Euler
angles system, may be extracted from equation (39) as:
T
I
= m ⋅ J ⋅ J
(40)
A E( eq)
A
E
E
T
V
= m ⋅ J J
⋅
(41)
A E( eq)
A
E
E
These matrices represent the inertia matrix and the Coriolis and centripetal terms matrix of a
virtual mobile platform that is equivalent to the six actuators.
3.3 Fixed-length links modeling
If the centre of mass of each fixed-length link, cmL, is considered to be located at a constant
distance bcm from the fixed-length link to mobile platform connecting point (Figure 4) then
its position relative to frame {B} is:
b
B
B
P
cm
p
= x
+ p
−
⋅ a
(42)
L
P
i
( pos)
i
i
B
B
B
L
Equation (42) may be successively rewritten as:
b
B
B
P
cm
p
= x
+ p
−
⋅ x
− b + p
− d
(43)
L
P pos
i
( B
P
P pos
i
i
i )
i
(
)
(
)
B
B
B
B
B
L
⎛
b ⎞
⎛
b ⎞
b
b
B
cm
B
cm
P
cm
cm
p
= ⎜1−
⋅⎟ x
+ ⎜1−
⋅⎟ p +
⋅ b +
⋅ d
(44)
L
P
i
( pos)
i
i
i
B
B
B
⎝
L ⎠
⎝
L ⎠
L
L
Dynamic Model of a 6-dof Parallel Manipulator Using the Generalized Momentum Approach
79
B p
being a vector expressed in frame {B}.
Li B
bcm
P
Pp
i
i
P
a i cmL
B x ( Ppo ) s B
B
d
p
i
Li B
z B
Bi
y
b
B
i
B
x B
Figure 4. Position of the centre of mass of a fixed-length link i
The linear velocity of the fixed-length link centre of mass, Bp
, relative to {B} and
Li B
expressed in the same frame may be computed from the time derivative of equation (44):
⎛
b ⎞
b
B
cm
p
= ⎜1−
⎟ ⋅ x
+ ω
× p
+
⋅ d
L
( B
B
P
P pos
P
i
) cm
i
(
)
i
B
B
B
B
⎝
L ⎠
L
(45)
⎛
b ⎞
b
cm
= ⎜1−
⎟ ⋅ ( B
B
P
x
+ ω
× p
+
⋅ l
⋅ z
P pos
P
i
) cm
(
)
i
B
B
B
B
⎝
L ⎠
L
Equation (45) can be rewritten as:
⎡ B v
⎤
B p
= J ⋅
(46)
L
B
⎢
P B
B
⎥
i
B
i
⎢⎣ ω P B ⎥⎦
where the jacobian J is given by:
i
B
⎡⎢ 1
0
0
⎛
bcm ⎞ ⎢
J =
B
⎜1−
⎟ ⋅
i
L
⎢
0
1
0
⎝
⎠ ⎢ b
b
b
cm
cm
cm
J
J
J
+1
⎢⎣ L −
Ci 1
b
L −
Ci 2
b
L −
Ci 3
b
cm
cm
cm
⎤
(47)
0
P p
− Pp
⎥
i
i
Bz
By
⎥
− Pp
0
P p
i
i
⎥
Bz
Bx
⎥
b
b
b
P p
+
cm
J
− Pp
+
cm
cm
J
J ⎥
i By
L −
Ci 4
i Bx
b
L −
Ci 5
b
L −
Ci 6
b
⎥
cm
cm
cm
⎦
being J Cij the elements of line i column j of matrix J C.
The linear momentum of each fixed-length link, Q
, can be represented in frame {B} as:
Li B
80
Parallel Manipulators, Towards New Applications
B
Q
= m ⋅ p
(48)
L
L
L
i
B
i
B
where mL is fixed-length link mass.
Introducing jacobian J and transformation T in the previous equation results into:
i
B
B
Q
= m ⋅ J ⋅ T⋅ x
(49)
L
L
B
i
i
P
B
B| E
The inertial component of the force applied to the fixed-length link due to its translation and
expressed in {B} can be obtained from the time derivative of equation (49):
d
Li f
= Q
= m ⋅
J ⋅ T ⋅ x
+ m ⋅ J ⋅ T⋅ x
(50)
L ine tra
L
L
( B ) B
B
i (
)( )
i