Apply fuzzy interpolation method to calibrate parallel

8 Pages·0.535 MB·English

by  Ying Bai, Hanqi Zhuang, Dali Wang.

IntJAdvManufTechnol
DOI10.1007/s00170-011-3613-y
ORIGINAL ARTICLE
Apply fuzzy interpolation method to calibrate parallel
machine tools
Ying Bai&Hanqi Zhuang&Dali Wang
Received:7May2009/Accepted:25August2011
#Springer-VerlagLondonLimited2011
Abstract A novel and efficient fuzzy interpolation method their applications [1–5]. PMTs have the advantage of great
is proposed to simplify the calibration process for parallel operational accuracy due to their high rigidity. However,
machine tools (PMTs). Either inverse or forward kinematic because of the nonlinear relationship that existed between
models must be used in the traditional PMT calibration the actuators and the platform, it is difficult to develop a
methods to perform the identification and compensation of forward kinematic model for PMTs. Most popular methods
the pose errors for PMTs, which made the calibration for kinematic analysis are iterative algorithms, which are
processtime consumingandinefficient inrealapplications. inefficient and may lead to unstable solutions.
Instead of using a model, the proposed method presents a DifferentcalibrationmethodsandstrategiesforPMTshave
modeless technique combined with the fuzzy interpolation beenreportedrecently[6–20]. However, these methods used
method to obtain high calibration accuracies when a small forward and inverse kinematic models to establish relation-
workspace is adopted. This new approach can significantly ships between the pose of the end-effector and the link
reducethecomplexintraditionalPMTcalibrationprocesses lengths as well as the joints on the mobile and base
and greatly simplify the calibration procedures. platforms. The complication in developing such forward or
inverse kinematic models makes those calibration processes
Keywords Parallel machine tools.Robots calibrations. very complicated, time consuming, and inefficient.
Fuzzy interpolations.Kinematic models Gengetal.developedsomeneuralnetworkalgorithmsto
derive an approximate forward kinematic model of PMTs
[21]. However, the algorithm is computationally intensive
1 Introduction and does not guarantee convergence.
Tosimplifythecalibrationprocess,amodelesscalibration
In recent years, parallel machine tools (PMTs) have drawn method with a fuzzy interpolation technique (FIM) is
more and more interest in the academia and industry for developedinthispaper.Themainadvantageoftheproposed
techniqueisthatitdoesnotinvolveanykinematicmodeling
and identification process. The calibration process can be
Y.Bai(*)
significantly simplified, and calibration performance can be
JohnsonC.SmithUniversity,
improvedbyusingtheproposedmethodaslongasthePMTs
Charlotte,NC,USA
e-mail:[email protected] workinarelativelysmallworkspace.
Compared with other interpolation techniques, the FIM
H.Zhuang
has the advantages of high calibration accuracy, shortened
FloridaAtlanticUniversity,
calibration process, and potentially low operational cost.
BocaRaton,FL,USA
e-mail:[email protected] One of the most prominent features of PMTs is that they
have relatively small workspace. This makes them good
D.Wang
candidates for the proposed technique.
ChristopherNewportUniversity,
The paper is organized in the following sequence: after
NewportNews,VA,USA
e-mail:[email protected] the introduction section, a description of the structure and
IntJAdvManufTechnol
kinematics of PMTs is given. The architecture and equationoftheparallelmachinetools.Foragiven setof li,
algorithm of FIM are provided in Section 3. Simulation a unique pose of the mobile platform {x y z α β γ} can be
results are discussed in Section 4, and the paper ends with determined. The goals of the calibration for the parallel
conclusions in Section 5. machinetools areto find the actual vector parameters ai, bi
and link length offset Δli of li and to perform the
compensations for the nominal poses of the mobile
2 Kinematics of parallel manipulators platform based on those identified parameters in order to
improve the pose accuracy of the parallel manipulators
The typical structure of a spatial six SPS parallel machine while they are in operation.
tool known as Stewart platform is shown in Fig. 1. Generally, it is difficult to derive a forward kinematic
Theframesofthemobileplatformandthebaseplatform model for parallel manipulators since there is no closed
are defined as {A} and {B}, respectively. The spherical formsolution,andsomenumericalalgorithmsmustbeused
joints connected to the mobile platform are indicated as Ai to find the “true” forward kinematic parameters [3–7].
andtheU-jointslocatedonthebaseplatformaredefinedas
Bi (i=1,2,…6). The position vectors of 6 joint points Bi in
thebaseplatformwithrespectto{B}aredenotedasb ∼b , 3 Fuzzy interpolation method
1 6
and the position vectors of the joint points A ∼A in the
1 6
mobile platform with respect to {A}are denoted as a ∼a . Generally, themodelesscalibration can bedivided into two
1 6
ThepositionvectoroftheoriginO of{A}withrespectto steps: measurement and compensation.
A
{B} is defined as t, and the rotation matrix of {A} respect
to {B} can be expressed as R that is a 3×3 rotation matrix 3.1 Measurement process
with a set of Euler angles [α, β, γ]T. The translation vector
t, and the rotation matrix R can be written as To begin, a measuring device is used to measure a set of
selected poses that are distributed around a group of grid
t ¼½xt;yt;zt(cid:2)T ð1Þ points (vertices) of all cubic cells in the PMTworkspace in
R¼Rðz;gÞRðy;bÞRðx;aÞ
ordertodeterminetheerrorsbetweenthedesiredposesand
the nominal poses of the end-effector. All the pose errors
2 3 are stored into computer memory based on each pose’s
cgcb cgsbsa(cid:3)sgca cgsbcaþsgsa
¼4sgcb sgsbsaþcgca sgsbca(cid:3)cgsa5 ð2Þ coordinates. During the calibration process, a target pose
canbeinterpolatedbyusingtheFIMbasedontheerrorsof
(cid:3)sb cbsa cbca
neighboringposeslocatedattheverticesofenclosingcubic
whereR(z,γ)isarotationofγaboutthez-axis(Roll),R(y,β) cell.Onebenefitofthistechniqueisthatitcancombinethe
is a rotation of β about the y-axis (Pitch), and R(x, α) is a identificationandcompensationprocessintoone.Thisgreatly
rotation of α about the x-axis (Yaw). reducesthetimeconsumedinthecompensationprocess.
The kinematic equation of the parallel machine tool can When implementing the modeless method to calibrate a
be expressed as: robot, it is necessary to use a measurement tool, such as a
high-accuracy coordinate measuring machine or a laser
li ¼jjRAaiþt(cid:3)Bbijjði¼1;2;...6Þ ð3Þ tracking system (LTS) to record pose errors as the robot
whereAairepresentstheithpositionvectoraiinthemobile moves its end-effector to all grid points on pre-determined
platform respecttothe{A}andBbidenotestheithposition cubic cells. Figure 2 shows an experiment setup in which
vector bi in the base platform respect to {B}. li is the ith
link length. Equation 3 is a typical inverse kinematic
Fig.1 Stewartplatform
Fig.2 Ameasurementsetupformodelessmethod
IntJAdvManufTechnol
an LTS is used to measure pose errors of a PMT. A similar are ex, ey, and ez. The outputs are eex, eey, eez, eeα, eeβ,
arrangement can be made for measuring pose errors of a and eeγ, which are shown in Fig. 3b.
robot. The relationship between the frame of the LTS {W} The control rule is shown in Fig. 3c, which is
and the Base of the PMT {B} is also shown in Fig. 2. straightforward and based on the human knowledge. It is
The PMTworkspace is divided into a sequence of small worth to note that each Pi should be considered as a
cubic cells. Each small cell is surrounded by eight combination of three position and three orientation error
neighboring grid points (vertices) as shown in Fig. 2. At components on each grid point.
each grid point, the LTS is used to measure the pose error Inourcurrentstudy,thedistancebetweentheneighboring
of the end-effector of the manipulator. In Fig. 2, the gridpointsofeachcellis20mminthreedirections,whichisa
nominalposeofthegridpointi is Pi=[xi, yi, zi, αi, βi, γi]T, standardintervalforsmallcalibrationworkspaces.
and the actual pose of the end-effector is Pia=[xia, yia, zia, In total, the workspace includes 20×20×20 cells, and it
αia, βia, γia]T. Thus, the pose error for this grid point is is equivalent to a 400×400×400-mm3 space. This is a
Pia−Pi. The parallel manipulator will be moved to all grid typical workspace of most parallel manipulators imple-
points in its workspace, and all pose errors on these grid mented in real operations [6, 7]. The input membership
points are measured and stored in the memory for future functions for x, y, and z directions, and predefined output
use. membership functions are shown in Fig. 4.
To measure the orientation of the end-effector, different Thepredefinedoutputmembershipfunctionsareusedas
methods [3, 10, 11, 22, 23] have been reported. In this thedefaultones,andtheactualoutputmembershipfunction
study, a similar method to [23] is used. A special retro- will be obtained by shifting the default one based on the
reflectorisinstalledontheend-effectorofthePMT,andthe actual error value on the grid points. For each cell, eight
orientation information of the end-effector can be obtained output membership functions are implemented and each
from a charged coupled device effectively [23]. oneisassociatedwiththeerroratonegridpoint.InFig.4b,
only four orientation output membership functions are
3.2 Compensation process shown because of space limitation. In a real application,
total eight orientation and eight position membership
During the compensation stage, as the manipulator moves functions should be utilized.
to a certain target point in its workspace, an interpolation The Gaussian-bell waveforms are selected as the shape
technique is used to estimate the target pose error based on of the membership functions for three inputs. As shown in
the errors of the neighboring grid points around the target Fig. 4a, the ranges of inputs are between −10 and 10 mm
point. Finally, this error is added into the target pose to (20-mm interval on grid points). The reason for this
obtainthecompensatedposewhichthePMTiscommanded selection is that the Gaussian-bell waveform has a smooth
moving to. curve and can improve the accuracy. In Fig. 4, Wand E
A dynamic online fuzzy interpolation algorithm is used represent the inputs located at different areas in the x
for the error interpolation process of this parallel manipu- direction, N and S represent those in the y direction, and L
lator’s calibration. Unlike traditional offline fuzzy inference and U represent those in the z direction. Unlike the
algorithms, in which both input and output membership traditionalfuzzyinferencesystem,inwhichallmembership
functionsaswellasthelookuptablehavebeendefinedinthe functionsshouldbedeterminedtoproducethelookuptable
designstagebasedontheinputandoutputranges,theoutput prior to the implementation of the fuzzy system, in this
membership functions and lookup table implemented in this study, the output membership functions will not be defined
applicationaredefinedbasedontherealposeerrorsmeasured until the implementation of the fuzzy error mapping to
during the real measurement process. The identification and compensate the pose errors. Thus, the output membership
compensation accuracy of the parallel manipulator can be functions will be determined during the application of the
significantly improved after using this algorithm since this fuzzy inference system online or dynamically. Figure 4b
algorithmusesthereal-timeposeerrorasinputstotheFIM. shows an example of the output membership functions,
The definition of this dynamic online fuzzy inference which are related to the simulated random errors at
algorithm is shown in Fig. 3. Each small cube, which is neighboring grid points. Each Pxi, Pyi, and Pzi corresponds
surroundedbyeightneighboringgridpoints,isdefinedasa to the pose error at the ith grid point, respectively. During
cell. Furthermore, this cell is divided equally into eight thedesignstage,alloutputmembershipfunctionsshouldbe
smaller cells, which are shown in Fig. 3a. initialized to a Gaussian waveform with a mean of 0 and a
TheposeerrorateachgridpointisdefinedasP ,P ,P , range that is closed to the actual possible output range
1 2 3
P , P , P , P , and P . For the fuzzy inference system, the which can be estimated based on the different parallel
4 5 6 7 8
interpolation method is divided into the three dimensions manipulators for the different applications. These output
separately, so the inputs to the fuzzy inference system membership functions will be determined online based on
IntJAdvManufTechnol
Fig.3 a–cDefinitionofthefuzzyinterpolationinferencesystem
the errors of the neighboring grid points around the target The control rules shown in Fig. 3c can be interpreted as
point in the workspace during the compensation process. followsaftertheoutputmembershipfunctionsaredetermined:
Ifex is W;eyis N;andez is U;theneexis Px1;eeyis Py1;and eezis Pz1; andeea isa1;eebis b1;andeeg is g1:ðP1Þ
Ifexis W;eyis N;and ezis L; then eexis Px3;eeyis Py3;andeez is Pz3;andeeaisa3;eeb is b3;andeegis g3:ðP3Þ
Ifex is W;eyis S;andezis U;theneexis Px5;eey is Py5;andeezis Pz5;andeea isa5;eeb isb5;andeegisg5:ðP5Þ
Ifex is W;ey is S;andez is L;theneexisPx7;eeyis Py7;andeezisPz7;andeea isa7;eebisb7 andeeg isg7:ðP7Þ ð4Þ
Ifex is E;ey is N; andez is U;theneex is Px2;eey isPy2;andeezis Pz2;andeeaisa2;eebisb2;andeegisg2:ðP2Þ
Ifexis E; ey is N; andezis L;theneexis Px4;eeyis Py4; andeezis Pz4;andeea isa4;eeb isb4;andeeg isg4:ðP4Þ
Ifex is E;eyis S;andez is U;theneex is Px6;eeyis Py6;andeez is Pz6;andeeaisa6;eebisb6;andeegisg6:ðP6Þ
Ifexis E;eyis S;andezis L; theneexis Px8;eeyis Py8; andeezis Pz8;andeeaisa8;eebisb8andeegisg8:ðP8Þ
Thecontrolrulesarestraightforward,andtheyarebased The input error variables can be expressed as a label set
on the human knowledge [24]. The error on P grid point L, with E being a linguistic input variable:
1
should carry larger weight if the target position (input) is
LðEÞ¼fNWU;NWL;NEU;NEL;SWU;SWL;SEU;SELg
located inside the NWU area on a cell. Similar consider-
ation should be given for errors on all other grid points. ð5Þ
Fig.4 a–cInputandoutputmembershipfunctions
IntJAdvManufTechnol
Assume that ui is the membership function, Ui the This shortcoming becomes of little importance as the
universe of discourse, and m the number of contributions, availability of high-speed CPUs for the controllers.
the traditional output of the fuzzy inference system can be
represented as:
4 Simulation results
Pm
ðui(cid:4)UiÞ
u¼i¼1 Pm ð6Þ To compare the performance of modeless and model-based
u calibration methods, a simulation study is provided hereaf-
i
i¼1 ter.AStewartplatformdevelopedintheRoboticsCenterat
where u is the current crisp output of the fuzzy inference Florida Atlantic University (FAU) is utilized to illustrate
this implementation. The simulation study is divided into
system and (6) is obtained by using the center-of-gravity
method. In this study, both ui and Ui in the output twoparts,measurement andcompensation process,aslisted
below. An LTS developed in the Robotics Center at FAU
membership functions are randomly distributed variables,
was used to perform the measurement task.
and the actual values of these variables depend upon the
A single-beam LTS is used for the tracking and
position errors of eight neighboring grid points around the
measurement purpose. The target is a retro-reflector
target position. These relationships can be expressed as:
installed on the end-effector of the PMT. The retro-
uxi ¼FxiðPx1;Px2;Px3;Px4;Px5;Px6;Px7;Px8Þ ð7Þ reflector is built with three prisms that can reflect both the
position and the orientation of the target [23].
Uxi ¼QxiðPx1;Px2;Px3;Px4;Px5;Px6;Px7;Px8Þ ð8Þ 4.1 Measurement Process Simulation Steps
where Fxi is the membership function of the input position
in the x direction, and it is a predetermined membership 1. For a set of given actual poses, using the inverse
function as shown in Fig. 4a. Qxi is the real error output kinematics (3) with identified kinematic parameters
membership function, which is a randomly distributed ra ¼½xai; yai; zai; xbi; ybi; zbi; Δli(cid:2)T[6] to obtain the
function, and it gives the error output contributions in the actual leg length vector l. These given poses can be
x direction.This membershipfunctionisdetermined bythe considered as the actual poses (identified poses) of the
realpositionerrorsatthe8-neighboringgridpointsinthex StewartplatformXa ¼½xyzabg(cid:2)Tateachgridpointof
direction, Px1∼Px8. This membership function determines the 3D workspace.
the degree to which the current position input belongs to 2. Add those actual poses given in step 1 with a uniform-
each different real error output based on the eight control distributed noise XaþArandðÞ to obtain the “actual”
rulesdefinedin(4)inthexdirection,anditisequivalentto poses Xm ¼½xyzabg(cid:2)T. These poses can be consid-
the universe of discourse or a weighing factor. Substituting
ered as the measured poses of the Stewart platform at
(7) and (8) into (6), one obtains:
each grid point of the 3D workspace.
3. Obtain the nominal poses Xn using the forward
Pm
FxiðPx1; Px2; :::::Px8Þ (cid:4)QxiðPx1; Px2;:::Px8Þ kinem(cid:2)aticmodel[1] with nomina(cid:3)l kinematic parameters
ui ¼ i¼1Pm ð9Þ rn ¼ xnai; ynai; znai; xnbi; ynbi; znbi; lin Tand the leg length
FxiðPx1; Px2; Px3; Px4;Px5;Px6;Px7;Px8Þ l obtained from step 1 as the input at each grid point
i¼1 of the 3D workspace.
Here, ux represents the final error output of the fuzzy 4. CalculatetheposeerrorsΔX ¼Xm(cid:3)Xn.Theseerrors
interpolationmethodinthexdirection.In(9),Qxiwillnotbe can be considered as the ones at grid points for each
determined until the fuzzy error interpolation technique is cell and should be stored in the memory to be used
appliedinanactualcompensationprocess,whichmeansthat during the compensation process.
this fuzzy inference system is an online process. The final
crisp output of the fuzzy error interpolation system is
4.2 Compensation Process Simulation Steps
determined by the neighboring pose errors of 8 grid points.
Similarcalculationscanbeimplementedfortheerroroutputs 5. Follow a step similar to step 1 to select some actual
in the y and z directions as well as in three orientations. poses Xa. The only difference is that those poses must
The advantage of using the online fuzzy inference not be located at the grid points. We can obtain the
system is its real-time control ability; the drawback is that actualleglengthvector lbyusingthose selectedactual
this method may result in a relative longer response time poses and the inverse kinematic model (3) combined
because of the calculation in the fuzzy inference system. with identified kinematic parameters
IntJAdvManufTechnol
Fig.5 Thesimulationresults 3
model based position errors
m) modeless position errors
m
s ( 2
or
err
n
sitio 1
o
P
0
1 2 3 4 5 6 7 8 9 10
0.015
d) model based orientation errors
a modeless orientation errors
s (r 0.01
or
err
n
o
ati 0.005
nt
e
Ori
0
1 2 3 4 5 6 7 8 9 10
The number of poses calibrated
ra ¼½xai; yai; zai; xbi; ybi; zbi; Δli(cid:2)T: 4.3 Details of Steps 3 through 7
6. Follow a step similar to step 2 above to obtain the
The presentation of forward kinematic models in parallel
measured or true poses Xm ¼½xyzabg(cid:2)Tby adding
machine tools is a very challenge topic since no closed
those actual poses with a uniform-distributed noise
form solution exists. A more detailed description about our
XaþArandðÞ:
handling for forward kinematics of the parallel machine
7. Obtain the nominal poses Xn using the forward
tools is provided in this section.
kinematicmodel[1] with nominal kinematic parameters
(cid:2) (cid:3) The relation between the pose and the kinematic
rn ¼ xnai;ynai;znai;xnbi;ynbi;znbi;lin T and the leg length parameters is:
l obtained from step 5 as the input at each non-grid
point of the 3D workspace.
8. UposseetehrerofruszΔzyX ifnotrertphoolsaetionnomminetahlopdosteoscXanlc,ualnatde atdhde ðliþΔliÞl2i ¼¼ðjRRaaiiþþtt(cid:3)(cid:3)bbiijÞTðRaiþt(cid:3)biÞ
calculated pose errors to those nominal poses Xn to get
the calibrated poses using the fuzzy interpolation
method, XF=Xn+ΔX. Table1 Modelessandmodel-basedmethodscomparison
9. Calculate the fuzzy interpolation pose errors by
Cubiccellsize Error Modeless Model-based
calculating the difference ΔXF=Xm−XF. Where Xm is type (mm) (mm)
obtained from step 6.
10. ComparethefuzzyinterpolatedposeerrorsΔXFthatcan 20×20×20mm3 Mean 0.0108 0.0092
beconsideredasthemodelesserrorswiththeposeerrors Max 0.0912 0.0875
ΔXobtainedinstep4byusingthemode-basedmethod. STD 0.0087 0.0152
11. Display the comparison between the two calibration 10×10×10mm3 Mean 0.0011 0.0012
errors. Max 0.0495 0.0513
STD 0.0020 0.0157
Steps 3 and 7 in the above simulation process need to
5×5×5mm3 Mean 0.0001 0.0010
use the forward kinematic model of the PMT to derive the
Max 0.0475 0.0500
poses based on the input leg lengths. Some details for the
STD 0.0002 0.0101
forward kinematic model are provided below.
IntJAdvManufTechnol
The cost function is using the proposed modeless method is better than that of
using a model-based method when workspaces are small.
X6 (cid:2) (cid:3) The model-based method can be replaced by the modeless
QðxÞ¼ lia(cid:3)jfiðx; uÞj 2 method as the PMTworks in a relative small workspace.
i¼1
Additional advantage of using a modeless method to
where fi(x, u) is the ith joint-link train inverse kinematic calibrate PMTs is that the kinematic modeling and
solutionforli,uisthekinematicparameters(ai,bi),andxis identification process can be removed, and furthermore
theposeofthePMT[25].Proceduretoperformtheforward the compensation process can be significantly simplified.
kinematics can be expressed by the following sequence: The modeless method is suitable for applications requiring
higher accuracy PMTs calibration.
1. Make an initial estimation of X for a given input, the
actual link length la
2. Perform inverse kinematics for each joint-link train to
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