B| E
dt
The inertial component of the generalized force applied to {P} due to each fixed-length link
rotation, Pf
, expressed using the Euler angles system, can be obtained pre-
Li ( ine)( rot ) B| E
multiplying equation (70) by matrix T T:
d
T P
T
T
T ⋅ f
= T ⋅ J ⋅
I
⋅ J ⋅ T ⋅ x
+
L ine rot
D
i
i
( L rot
D
i
i
) B
( )( )
( )
P
B
B
B| E
dt
(71)
T
T
B
T ⋅ J ⋅ I
⋅ J ⋅ T⋅ x
Di
Li ( rot )
Di
P
B
B| E
Dynamic Model of a 6-dof Parallel Manipulator Using the Generalized Momentum Approach
83
P
T P
f
= T ⋅ f
(72)
Li ( ine)( rot )
Li ( ine)( rot )
B| E
B
The inertia matrix and the Coriolis and centripetal terms matrix of the rotating fixed-length
link, expressed in the Euler angles system, may be extracted from equation (71) as:
I
T
T
(73)
i
L ( rot )
= T ⋅ J ⋅ I
⋅ J ⋅
E ( eq)
i
D
i
L ( rot )
T
i
D
B
d
V
= T T ⋅ J T ⋅
(74)
L rot
D
(I
⋅ J ⋅ T
L rot
D
)
i (
) E( eq)
i
i (
)
i
B
dt
These matrices represent the inertia matrix and the Coriolis and centripetal terms matrix of a
virtual mobile platform that is equivalent to each rotating fixed-length link.
It should be noticed that equations (24), (38), (51) and (69) by providing expressions for the
inertial component of the generalized force applied to {P} and expressed in {B} enable a clear
physical meaning to the moments applied to {P}.
3.4 Gravitational component of the RCID dynamic model
Given a general frame {x, y, z}, with z ≡ −gˆ , the potential energy of a rigid body is given by: P = m ⋅ g ⋅ z
(75)
c
c
c
where mc is the body mass, g is the modulus of the gravitational acceleration and z c the
distance, along z, from the frame origin to the body the centre of mass.
The gravitational components of the generalized forces acting on {P} can be easily obtained
from the potential energy of the different bodies that compose the system:
P
∂ ( B x
P
P
)
P
B| E
f
=
(76)
P( gra )
B
B| E
∂ x P B| E
P
∂ ( B x
A
P
)
P
i
B| E
f
=
(77)
Ai ( gra)
B
B| E
∂ x P B| E
P
∂ ( B x
L
P
)
P
i
B| E
f
=
(78)
Li ( gra)
B
B| E
∂ x P B| E
Vectors P f
, P f
and P f
represent the gravitational components of the
P( gra )
A
Li ( gra)
i ( gra )
B| E
B| E
B| E
generalized forces acting on {P}, expressed using the Euler angles system, due to, in that
order, the mobile platform, each actuator and each fixed-length link. Therefore, to be added
to the inertial force components, these vectors must be transformed, to be expressed in
frame {B}. This may be done pre-multiplying the gravitational components force vectors by
the following matrix:
⎡ℑ
0 ⎤
⎢
(79)
− T ⎥
⎣0 J A ⎦
84
Parallel Manipulators, Towards New Applications
4. Computational effort of the RCID dynamic model
The computational effort of the RCID dynamic model obtained through the use of the
generalized momentum approach is compared with the one resulting from applying the
Lagrange method using the Koditschek representation (Lebret et al., 1993; Koditschek, 1984).
As the largest difference between the two methods rests on how the Coriolis and centripetal
terms matrices are calculated, the two models are evaluated by the number of arithmetic
operations involved in the computation of these matrices. The results were obtained using
the symbolic computational software Maple® and are presented in Table 2.
Generalized
Lagrange
Momentum
Add. Mult. Div. Add. Mult. Div.
Mobile platform
310
590
0
94
226
0
Six actuators
3028 4403
30
724
940
18
Translating link
751
1579
6
131
279
4
Rotating link
2180 3711
7
355
664
7
Total operations 20924 36733 108 3734 6824
84
Table 2. Number of arithmetic operations involved in the computation of the Coriolis and
centripetal terms matrices of the RCID dynamic model
The dynamic model obtained by the generalized momentum approach is computationally
much more efficient and its superiority manifests precisely in the computation of the
matrices requiring the largest relative computational effort: the Coriolis and centripetal
terms matrices.
The proposed approach was used in the dynamic modelling of a 6-dof parallel manipulator
similar to a Stewart platform. Nevertheless, it can be applied to any mechanism.
5. Conclusion
Dynamic modelling of parallel manipulators presents an inherent complexity. Despite the
intensive study in this topic of robotics, mostly conducted in the last two decades, additional
research still has to be done in this area.
In this paper an approach based on the manipulator generalized momentum is explored and
applied to the dynamic modelling of parallel manipulators. The generalized momentum is
used to compute the inertial component of the generalized force acting on the mobile
platform. Each manipulator rigid body may be considered and analyzed independently.
Analytic expressions for the rigid bodies’ inertia and Coriolis and centripetal matrices are
obtained, which can be added, as they are expressed in the same frame. Having these
matrices, the inertial component of the generalized force acting on the mobile platform may
be easily computed. This component can be added to the gravitational part of the
generalized force, which is obtained through the manipulator potential energy.
The proposed approach is completely general and can be used as a dynamic modelling tool
applicable to any mechanism.
The obtained dynamic model was found to be computationally much more efficient than the
one resulting from applying the Lagrange method using the Koditschek representation. Its
superiority manifesting precisely in the computation of the matrices requiring the largest
relative computational effort: the Coriolis and centripetal terms matrices.
Dynamic Model of a 6-dof Parallel Manipulator Using the Generalized Momentum Approach
85
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5
Redundant Actuation
of Parallel Manipulators
Andreas Müller
Institute of Mechatronics at the Chemnitz University of Technology
Germany
1. Introduction
High stiffness, low inertia, large accelerations, and high precision are desirable properties
attributed to parallel kinematics machines (PKM). However, relatively small workspace and
the abundance of singularities within the workspace partly annihilate the aforementioned
advantages. Redundant actuation and novel redundant kinematics are means to tackle these
shortcomings. Redundant parallel kinematics machines are ideal candidates for use in high-
precision applications, such as robot-assisted surgery. Their advantageous features promise
to deliver the needed accuracy, stiffness, dexterity and reliability. Redundant actuation
admits to eliminate singularities, increase the usable workspace, augment the dexterity, and
partially control the internal forces. Actuator redundancy is also a means to improve fault
tolerance, as redundant actuators can compensate the failure of other actuators. Redundant
actuation increases the payload and acceleration, can yield an optimal load distribution
among the actuators, or can reduce the power consumption of the individual drives.
Actuator redundancy can also improve the force transmission properties and the
manipulator stiffness. It can be purposefully exploited for secondary tasks, such as the
generation of internal prestress and the generation of a desired compliance of the PKM. The
first can be used to avoid backlash, whereas the second admits to homogenize the stiffness
properties within the workspace. Kinematically redundant PKM, i.e. systems that possess a
higher mobility then required for the task, allow to circumvent singularities as well as
obstacles, and to increase the dexterity.
The control of redundantly actuated PKM poses additional challenges, rooted in the
resolution of the redundancy within the control schemes. Whereas, model-based control
techniques can be directly applied to the control of non-redundantly actuated PKM,
redundancy, however, brings up two specific problems, one is the computationally efficient
resolution of the actuation redundancy, and the other is the occurrence of unintentional
antagonistic actuation due to model uncertainties.
This chapter is devoted to the modeling and control of redundantly actuated PKM. The aim
of the chapter is to summarize concepts for dynamic modeling of redundantly actuated
PKM, with emphasize on the inverse dynamics and control, and to clarify the terminology
used in the context of redundant actuation. Based on a mathematical model, PKM are
regarded as non-linear control systems.
The chapter is organized as follows. A short literature review in section 2 is meant to
familiarize the reader with current developments and research directions. In order to point
out the potential of redundantly actuated PKM, a motivating example is given in section 3.