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機0405 11號 馬吟川 指導(dǎo)老師：許寶杰
THE DESIGN OF PARALLEL KINEMATIC MACHINE
TOOLS USING KINETOSTATIC PERFORMANCE
CRITERIA
http://arxiv.org/ftp/arxiv/papers/0705/0705.1038.pdf
1. INTRODUCTION
Most industrial machine tools have a serial kinematic architecture, which means that
each axis has to carry the following one, including its actuators and joints. High Speed
Machining highlights some drawbacks of such architectures: heavy moving parts require
from the machine structure high stiffness to limit bending problems that lower the
machine accuracy, and limit the dynamic performances of the feed axes.
That is why PKMs attract more and more researchers and companies, because they
are claimed to offer several advantages over their serial counterparts, like high structural
rigidity and high dynamic capacities. Indeed, the parallel kinematic arrangement of the
links provides higher stiffness and lower moving masses that reduce inertia effects. Thus,
PKMs have better dynamic performances. However, the design of a parallel kinematic
machine tool (PKMT) is a hard task that requires further research studies before wide
industrial use can be expected.
Many criteria need to be taken into account in the design of a PKMT. We pay special
attention to the description of kinetostatic criteria that rely on the conditioning of the
Jacobian matrix of the mechanism. The organisation of this paper is as follows: next
section introduces general remarks about PKMs, then is explained why PKMs can be
interesting alternative machine tool designs. Then are presented existing PKMTs. An
application to the design of a small-scale machine tool prototype developed at IRCCyN
is presented at the end of this paper.
2. ABOUT PARALLEL KINEMATIC MACHINES
2.1. General Remarks
The first industrial application of PKMs was the Gough platform (Figure 1),
designed in 1957 to test tyres1. PKMs have then been used for many years in flight
simulators and robotic applications2 because of their low moving mass and high dynamic
performances. Since the development of high speed machining, PKMTs have become
interesting alternative machine tool designs3, 4.
Figure 1. The Gough platform
In a PKM, the tool is connected to the base through several kinematic chains or legs
that are mounted in parallel. The legs are generally made of either telescopic struts with
fixed node points (Figure 2a), or fixed length struts with gliding node points (Figure 2b).
Along with high-speed cutting's unceasing development, the traditional tandem type organization constructs the platform the structure rigidity and the traveling carriage high speed becomes the technological development gradually the bottleneck, but the parallel platform then becomes the best candidate object, but was opposite in the tandem engine bed, the parallel working platform had the following characteristic and the merit:
(1) structure is simple, the price is low The engine bed mechanical spare part number is series connected constructs the platform to reduce largely, mainly by the ball bearing guide screw, the Hooke articulation, the ball articulation, the servo electrical machinery and so on common module is composed, these common modules may by the special factory production, thus this engine bed's manufacture and the inventory cost are much lower than the same function's traditional engine bed, easy to assemble and the transporting.
(2) structure rigidity is high Because used closeness structure (closed-loop structure) to enable it to have high rigid and the high speed merit, its structural load streamline was short, but shouldered decomposes pulls, the pressure also to withstand by six connecting rods, by materials mechanics' viewpoint, when the external force was certain, the bracket quantity's stress and the distortion were biggest, the both sides inserted the (build-in) next best, came is again both sides Jan supports (simply-supported), next was the bearing two strength structure, what the stress and the distortion were smallest was the tensity two strength structure, therefore it had the high rigidity. Its rigidity load ratio is higher than traditional the numerically-controlled machine tool.
(3) processing speed is high, the inertia is low If the structure withstands the strength will change the direction, (will be situated between tensity and pressure), two strength components will be most save the material the structure, but it will move to the moving parts weight to reduce to lowly and simultaneously will actuate by six actuating units, therefore machine very easy high speed, and will have the low inertia.
(4) working accuracy is high Because it for the multiple spindle parallel organization composition, six expandable pole poles long alone has an effect to cutting tool's position and the posture, thus does not have the traditional engine bed (i.e. connects engine bed) the geometrical error accumulation and the enlargement phenomenon, even also has the being uniform effect (averaging effect); It has the hot symmetrical structural design, therefore the thermal deformation is small; Therefore it has the high accuracy merit.
(5) multi-purpose flexible Is convenient as a result of this engine bed organization simple control, easily according to processing object, but designs it the special purpose machine, simultaneously may also develop the general engine bed, with realizes the milling, boring, processings and so on grinding, but may also provide the essential measuring tool to compose it the measuring engine, realizes engine bed's multi-purpose. This will bring the very big application and the market prospect, has the very broad application prospect in the national defense and the civil aspect.
(6) service life is long Because the stress structure is reasonable, the moving part attrition is small, and does not have the guide rail, does not have the iron filings either the refrigerant enters the guide rail interior to cause it to scratch, the attrition or the corrosion phenomenon.
(7) Stewart platform suits in the modular production Regarding the different machine scope, only need change the connecting rod length and the contact position, maintains also easily, does not need to carry on part's remaking and to adjust, only need the new organization parameter input.
(8) transformation coordinate system is convenient Because does not have the entity coordinate system, the engine bed coordinate system and the work piece coordinate system transform depend on the software to complete completely, is convenient.
When the Stewart platform applies in the engine bed and the robot, may reduce the static error (, because high rigidity), as well as dynamic error (because low inertia). But Stewart the platform inferiority lies in its working space to be small, and it has the singular point limit in the working space, but the serial operation platform, the controller meets time the singular point, accountant will figure out the actuation order which the drive is unable to achieve to create the ning error, but the Stewart platform will lose the support partial directions in the strange position the strength or moment of force ability, will be unable to complete the constant load object.
Figure 2a. A bipod PKM
Figure 2b. A biglide PKM
2.2. Singularities
The singular configurations (also called singularities) of a PKM may appear inside
the workspace or at its boundaries. There are two types of singularities5. A configuration
where a finite tool velocity requires infinite joint rates is called a serial singularity. A
configuration where the tool cannot resist any effort and in turn, becomes uncontrollable,
is called a parallel singularity. Parallel singularities are particularly undesirable because
they induce the following problems:
- a high increase in forces in joints and links, that may damage the structure,
- a decrease of the mechanism stiffness that can lead to uncontrolled motions of the
tool though actuated joints are locked.
Figures 3a and 3b show the singularities for the biglide mechanism of Fig. 2b. In
Fig. 3a, we have a serial singularity. The velocity amplification factor along the vertical
direction is null and the force amplification factor is infinite.
Figure 3b shows a parallel singularity. The velocity amplification factor is infinite
along the vertical direction and the force amplification factor is close to zero. Note that a
high velocity amplification factor is not necessarily desirable because the actuator
encoder resolution is amplified and thus the accuracy is lower.
Figure 3a. A serial singularity
Figure 3b. A parallel singularity
2.3. Working and Assembly Modes
A serial (resp. parallel) singularity is associated with a change of working mode6
(resp. of assembly mode). For example, the biglide has four possible working modes for
a given tool position (each leg node point can be to the left or to the right of the
intermediate position corresponding to the serial singularity, Fig. 4a) and two assembly
modes for a given actuator joint input (the tool is above or below the horizontal line
corresponding to the parallel singularity, Fig. 4b). The choice of the assembly mode and
of the working mode may influence significantly the behaviour of the mechanism5.
Figure 4a. The four working modes
Figure 4b. The two assembly modes
3. PKMs AS ALTERNATIVE MACHINE TOOL DESIGNS
3.1. Limitations of Serial Machine Tools
Today, newly designed machine tools benefit from technological improvements of
components such as spindles, linear actuators, bearings. Most machine tools are based on
a serial architecture (Figure 5), whose advantage is that input/output relations are simple.
Nevertheless, heavy masses to be carried and moved by each axis limit the dynamic
performances, like feed rates or accelerations. That is why machine tools manufacturers
have started being interested into PKMs since 1990.
3.2. PKMs Potentialities for Machine Tool Design
The low moving mass of PKMs and their good stiffness allow high feed rates (up to
100 m/min) and accelerations (from 1 to 5g), which are the performances required by
High Speed Machining.
PKMs are said to be very accurate, which is not true in every case4, but another
advantage is that the struts only work in traction or compression. However, there are
many structural differences between serial and parallel machine tools, which makes it
hard to strictly compare their performances.
3.3. Problems with PKMs
a) The workspace of a PKM has not a simple geometric shape, and its functional
volume is reduced, compared to the space occupied by the machine7, as we can see on
Fig. 5
Figure 5. Workspace sections of Tricept 805
b) For a serial mechanism, the velocity and force transmission ratios are constant in
the workspace. For a parallel mechanism, in contrast, these ratios may vary significantly
in the workspace because the displacement of the tool is not linearly related to the
displacement of the actuators. In some parts of the workspace, the maximal velocities
and forces measured at the tool may differ significantly from the maximal velocities and
forces that the actuators can produce. This is particularly true in the vicinity of THE DESIGN OF PKMT USING KINETOSTATIC PERFORMANCE CRITERIA 5
singularities. At a singularity, the velocity, accuracy and force ratios reach extreme
values.
c) Calibration of PKMs is quite complicated because of kinematic models
complexity8.
4. EXISTING PKMT DESIGNS
In this section will be presented some existing PKMTs.
4.1. Fully Parallel Machine Tools
What we call fully parallel machine tools are PKMs that have as many degrees of
freedom as struts. On Fig. 7, we can see a 3-RPR fully parallel mechanism with three
struts. Each strut is made of a revolute joint, a prismatic actuated joint and a revolute
joint.
Figure 6. 3-RPR fully parallel mechanism
Fully PKMT with six variable length struts are called hexapods. Hexapods are
inspired by the Gough Platform. The first PKMT was the hexapod “Variax” from
Giddings and Lewis presented in 1994 at the IMTS in Chicago. Hexapods have six
degrees of freedom. One more recent example is the CMW300, a hexapod head designed
by the Compagnie Mécanique des Vosges (Figure 7)
.
Figure 7. Hexapod CMW 300 (perso.wanadoo.fr/cmw.meca.6x/6AXES.htm)
Fully parallel machine tools with fixed length struts can have three, four or six legs.
The Urane SX (Figures8 and 13) from Renault Automation is a three leg machine,
whose tool can only move along X, Y and Z axes, and its architecture is inspired from
the Delta robot9, designed for pick and place applications. The Hexa M from Toyoda is a
PKMT with six fixed length struts (Figure 9).
Figure 8. Renault automation Urane SX (from “Renault
Automation Magazine”, n° 21, may 1999)
Figure 9. Toyoda Hexa M (www.toyodakouki.
co.jp)
4.2. Other Kinds of PKMT
The Tricept 805 is a widely used PKMT with three variable length struts (Figures 5
and 10). The Tricept 805 has a hybrid architecture: a two degrees of freedom wrist
serially mounted on a tripod architecture.
Another non fully parallel MT is the Eclipse (Figure 11) from Sena Technology10, 11.
The Eclipse is an overactuated PKM for rapid machining, capable of simultaneous five
faces milling, as well as turning, thanks to the second spindle.
Figure 10. Tricept 805 from Neos robotics
(www.neosrobotics.com)
Figure 11. The Eclipse, from Sena Technology
(macea.snu.ac.kr/eclipse/homepage.html)
5. DESIGNING A PKMT
5.1. A Global Task
Given a set of needs, the most adequate machine will be designed through a set of
design parameters like the machine morphology (serial, parallel or hybrid kinematic
structure), the machine geometry (link dimensions, joint orientation and joint ranges), the
type of actuators (linear or rotative motor), the type of joints (prismatic or revolute), the
number and the type of degrees of freedom, the task for which the machine is designed.
These parameters must be defined using relevant design criteria.
5.2. Kinetostatic Performance Criteria are Adequate for the Design of PKMTs
The only way to cope with problems due to singularities is to integrate kinetostatic
performance criteria in the design process of a PKMT. Kinetostatic performance criteria
evaluate the ability of a mechanism to transmit forces or velocities from the actuators to
the tool. These kinetostatic performance criteria must be able to guaranty minimum
stiffness, accuracy and velocity performances along every direction throughout the
workspace of the PKMT.
To reach this goal, we use two complementary criteria: the conditioning of the
Jacobian matrix J of the PKMT, called conditioning index, and the manipulability
ellipsoid associated with J12. The Jacobian matrix J relates the joint rates to the tool
velocities. It also relates the static tool efforts to the actuator efforts. The conditioning
index is defined as the ratio between the highest and the smallest eigenvalue of J. The
conditioning index varies from 1 to infinity. At a singularity, the index is infinity. It is 1
at another special configuration called isotropic configuration. At this configuration, the
tool velocity and stiffness are equal in all directions. The conditioning index measures
the uniformity of the distribution of the velocities and efforts around one given
configuration but it does not inform about the magnitude of the velocity amplification or
effort factors.
The manipulability ellipsoid is defined from the matrix (J JT)-1. The principal axes of
the ellipsoid are defined by the eigenvectors of (J JT)-1 and the lengths of the principal
axes are the square roots of the eigenvalues of (J JT)-1. The eigenvalues are associated
with the velocity (or force) amplification factors along the principal axes of the
manipulability ellipsoid.
These criteria are used in Wenger13, to optimize the workspace shape and the
performances uniformity of the Orthoglide, a three degree of freedom PKM dedicated to
milling applications (Figure 12).
Figure 12. A section of Orthoglide’s optimised workspace
5.3. Technical Problems
If the struts of the PKMT are made with ballscrews, the PKMT accuracy may suffer
from struts warping due to heating caused by frictions generated by ballscrews. This
problem is met by hexapods designers that use ballscrews. Thus, besides manufacturing
inaccuracies, the calibration of a PKMT will have to take into account dimensions
variations due to dilatation. A good thermal evacuation can minimise the effects of
heating.
In case PKMT actuators are linear actuators, magnetic pollution has to be taken into
account so that chips clearing out is not obstructed. One technique, used by Renault
Automation for the Urane SX, is to isolate the tool from the mechanism.
At last, choosing fixed length or variable length struts influence the behaviour of the
machine. Actuators have to be mounted on the struts in case of variable length struts,
which increases moved masses. Fixed length struts do not have this problem, and
furthermore allow the use of linear actuators, that bring high dynamic performances.
6. CONCLUSIONS
The aim of this article was to introduce a few criteria for the design of PKMTs, which
may become interesting alternatives for High Speed Machining, especially in the milling
of large parts made of hard material, or for serial manufacturing operations on
aeronautical parts.
Kinetostatic criteria seem to be well adapted to the design of PKMTs, particularly for
the kinematic design and for the optimisation of the workspace shape, with regard to
performances uniformity.
The kinetostatic criteria have been used for the design of the Orthoglide, a three-axis
PKMT developed at IRCCyN. A small scale prototype is under development. A five-axis
PKMT will be derived from the Orthoglide.
11