(s)
sols #
sols #
(kbytes)
MSSM
octahedron
T,P
T,P
0,07
16
16
60
TSSM
hexapod
S,P
T,P
0,08
16
16
76
SSM
(6|)
S,P
S,P
0,67
36
16
238
6-6pp
(6|)
P
P
1,1
40
16
390
6-6p
(6|)
NP
P
1,8
40
16
308
6-6
(6|)
NP
NP
10,4
40
16
402
DIET
(6|)
NP
NP
9,9
40
40
392
Hexa
6R-6R-6R
P
S,P
2,0
40
8
346
Hexaglide
6T-6R-6R
P
S,P
0,5
36
8
180
Table 5. Hexapod FKP overall results and performances
6.6.3 Discussion on the results
The gradual passage to geometrically simpler parallel structures leads to significantly
shorter response times. In the 6-6 case, the introduction of one planar platform reduces
computation times by 20. Inasmuch, the passage from a theoretical SSM to a realistic 6-6
leads to computation times which are 40 times longer.
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201
With model simplification, it is notable that the real solution number is maintained. One
could conjecture homothopy in those cases. The number of complex roots varies when the
manipulator type changes; the MSSM and TSSM only have 16 solutions whereas the SSM
has 36. This last result raises an important classification issue which was not formerly
identified, [Faugère and Lazard 1995, Merlet 1997]. The SSM manipulator does not go in the
class of 6-6 manipulators and becomes a class by itself.
The table second section comprises FKP solving results about the hexapod obtained with the
Dietmaier's method, [Dietmaier 1998], the Hexa and the Hexaglide with a SSM type mobile
platform. In all instances, the Hexa, [Pierrot et al. 1991], and Hexaglide, [Hebsacker 1998]
can feature either 36 or 40 complex solutions, this only depends on the mobile platform
configuration. In fact, when the truncated equilateral triangle is used, then the FKP yield 36
solutions.
Knowing that the 6-6pp and 6-6p lead to 40 complex solutions, the new SSM class is
characterized not only by its complex solution number but by the mobile platform and fixed
base peculiar type.
Therefore, the manipulators which fall into the SSM category are the ones which feature one
specific truncated platform.
Although Dietmaier's 6-6 FKP yields the largest number of real solutions, it does not
necessarily lead to the longest computation times and largest result files.
Finally, various tests were performed where leg lengths were changed such as moving the
robot on a straight line or circles with the same 6-6 manipulator configuration. The real root
numbers have all been an even number in the set {4,8,12,16} depending on the location
inside the workspace. This could be considered a conjecture. The only case where only one
real solution has been found is when a theoretical 6-6pp has its actuator values bringing the
manipulator mobile platform to lie on the fixed base plane. This solution corresponds to two
real coincident roots and this case can be identified as a singularity with the loss of three
DOF since the manipulator can then only move in a plane.
6.7 Assembly mode analyses
The exact method allows addressing the question of assembly modes. This problem is also
referred to as posture analysis. Assembly modes are defined as follows:
Fig. 12. Posture analysis for the general 6-6 manipulator
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Parallel Manipulators, Towards New Applications
Definition 6.1 Given a manipulator configuration where the fixed base, the mobile platform and
kinematics chain lengths are specified, for a set of active joint positions, determine all the possible
geometric assemblies for the selected manipulator.
For the selected example, eight real solutions were obtained which geometrically represent
the only possible assembly modes. These modes are then drafted using the XMuPAD
environment, fig. 12. The position based models lead to root results which are directly
usable to draw the effective postures. This exemplifies that posture analysis is feasible for
any manipulator which can be modelled as a general 6-6 hexapod.
7. Conclusion
In this chapter, one complete exact method to solve the parallel manipulator FKP has been
explained. The method was applied to the 6-6 general parallel manipulator, which is
recognized to yield the most difficult problem, and also to various other manipulators such
as the SSM, TSSM, MSSM, Hexa and Hexaglide.
Moreover, the modeling of the FKP was investigated. Six displacement based models and
two position-based models were derived for the 6-6 general parallel manipulator.
One complete algebraic method to solve the Forward Kinematics Problem was applied.
Although many methods can find solutions to some of these FKP systems, the proposed
algebraic exact method insures the exactness of the real solution results, since it is based on
one Gröbner Basis, which completely describes the ideal related to the original system. From
this basis, one computes the Rational Univariate Representation including one univariate
equation for root isolation. The selected algorithms always succeeded to solve any parallel
robot FKP in all tested non-singular instances.
The selected manipulator was a typical 6-6 hexapod known as the Gough platform in a
calibrated configuration, measured on a real parallel robot prototype constructed from a
theoretically singularity free SSM design.
The 8 polynomial formulations were implemented and compared. We identified three
models that allowed for computation termination, out of which two were retained since
their computations occur with acceptable performances: a displacement based formulation
with the rotation matrix Gröbner Basis with end-effector position and rotation matrix
parameters as variables and a formulation with three points on a platform.
Solving typical posture examples, the Rational Univariate Representation comprised a
univariate equation of degree 40 and 8 to 12 real solutions were computed depending on the
position in the workspace. The total computation averaged 130 seconds on a relatively old
computer. On a faster computer, the response time falls to less than one second. Hence, this
method can be suitable for small-scale trajectory pursuit applications.
This result is very important since any Gröbner Basis completely describes the mathematical
object related to the original system. From this basis, one can try to build an exact equivalent
system with the original one including one univariate equation. Up to author’s knowledge,
this is actually the only known method that is certified to establish a truly equivalent system
that preserves the properties.
Further testing led to favor the first formulation, since it yields slightly faster computations
and it gives directly the end-effector position. The quaternion-based models can lead to
difficulties in the case of simpler configurations and thus longer computation times, since it
is doubling the solution number, which is explained by the Rational Univariate Representation
Certified Solving and Synthesis on Modeling of the Kinematics. Problems of Gough-Type
Parallel Manipulators with an Exact Algebraic Method
203
equations having a degree twice as large as the others. Moreover, one final advantage is that
the displacement-based equations can be applied on any manipulator mobile platform.
8. Acknowledgment
I would like to thank my wife Clotilde for the time spent on rewriting and correcting the
book chapter in Word.
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10
Advanced Synthesis of the DELTA Parallel
Robot for a Specified Workspace
M.A. Laribi1, L. Romdhane1* and S. Zeghloul2
Laboratoire de Génie Mécanique, LAB-MA-05
Ecole Nationale d’Ingénieurs de Sousse, Sousse 40031,
Laboratoire de Mécanique des Solides,UMR 6610
Bd Pierre et Marie Curie, BP 30179,Futuroscope 86962 Chasseneuil2
Tunisia1,
France2
1. Introduction
Parallel manipulators have numerous advantages in comparison with serial manipulators:
Higher stiffness, and connected with that a lower mass of links, the possibility of
transporting heavier loads, and higher accuracy. The main drawback is, however, a smaller
workspace. Hence, there exists an interest for the research concerning the workspace of
manipulators.
Parallel architectures have the end-effector (platform) connected to the frame (base) through
a number of kinematic chains (legs). Their kinematic analysis is often difficult to address.
The analysis of this type of mechanisms has been the focus of much recent research. Stewart
presented his platform in 1965 [1]. Since then, several authors [2],[3] have proposed a large
variety of designs.
The interest for parallel manipulators (PM) arises from the fact that they exhibit high
stiffness in nearly all configurations and a high dynamic performance. Recently, there is a
growing tendency to focus on parallel manipulators with 3 translational DOF [4, 5, 8, 9, 10,
11, 12, 13,]. In the case of the three translational parallel manipulators, the mobile platform
can only translate with respect to the base. The DELTA robot (see figure 1) is one of the most
famous translational parallel manipulators [5,6,7]. However, as most of the authors
mentioned above have pointed out, the major drawback of parallel manipulators is their
limited workspace. Gosselin [14], separated the workspace, which is a six dimensional
space, in two parts : positioning and orientation workspace. He studied only the positioning
workspace, i.e. , the region of the three dimensional Cartesian space that can be attained by a
point on the top platform when its orientation is given. A number of authors have described
the workspace of a parallel mechanism by discretizing the Cartesian workspace. Concerning
the orientation workspace, Romdhane [15] was the first to address the problem of its
determination. In the case of 3-Translational DOF manipulators, the workspace is limited to
* Corresponding author. email :lotfi.romdhane@enim.rnu.tn
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Parallel Manipulators, Towards New Applications
a region of the three dimensional Cartesian space that can be attained by a point on the
mobile platform.
Fig. 1: DELTA Robot (Clavel R. 1986)
A more challenging problem is designing a parallel manipulator fo