When a machine tool is mounted at the tip of a robotic manipulator, the manipulator becomes more flexible (the natural frequencies are lowered). Moreover, for a given flexible manipulator, its compliance will be different depending on feedback gains, configurations, and direction of interest. Here, the compliance of a manipulator is derived analytically, and its magnitude is represented as a compliance ellipsoid. Then, using a two-link flexible manipulator with an abrasive cut off saw, the experimental investigation shows that the chattering varies with the saw cutting angle due to different compliance. The main work is devoted to finding a desirable cutting angle which reduces the chattering

Book, W.

Huggins, J.

Lew, J.

Magee, D.

English

NASA Technical Reports Server

N91-30529
EXPERIMENTAL INVESTIGATIONS OF THE EFFECTS OF CUTTING ANGLE
ON CHATTERING OF A FLEXIBLE MANIPULATOR
J. Lew, J. Huggins, D. Magee and W Book
FlexibleAutomation Laboratory
George W. Woodruff School of Mechanical Engineering
Georgia InstituteofTechnology
Atlanta, Georgia 30332
ABSTRACT
When a m,mnbinet.oolis mounted at the b'pofa roboticmanipu-
lator, tlJe _In/p_l/ator becomes mote/]exible (the natured frequencies
are lowered). Moreover, [or a given flexible manipulator, its compS-
ante will be different depending on feedback gains, configurations, and
directio_ of interest. In this paper, the compliance of a manipulator is
derived analytically, and it$ magnitude is represented aa a compliance
elIipeffd. Then, using a two rink tTe_ble manipulator _'th an abraMve
cut-off saw, the experimental investigation shown that the chattering
varies with the _w cutting angle due to the different compliance. The
main work is de_vted to finding a desirable cutting angle which reduces
the chattering.
1. INTRODUCTION
1.1 Motivation
In real world applications, robot manipulators are mechan-
ically very rigid by design. This rigidity is necessary for high
positioning accuracy; however, it becomes difficult to perform
operations when a rigid manipulator contacts a workpiece. On
the other hand, flexible manipulators can provide passive com-
pliance due to their link flexibility. With this structural com-
pliance, certain applications such as cutting a workpiece can
be performed with pure position control. Thus, the compliance
can provide a simple, inexpensive solution for certain applica-
tions that otherwise could not be achieved with position control
alone [1,2].
Figure 1 shows the block diagrams of the overall architec-
ture of a cutting process with pure position control. An abrasive
cut-off saw is mounted at the tip of a manipulator. Its link flex-
ibility is represented by a spring constant (KL). The position
feedback signal is measured at each joint. Due to the flexibility
of the link, it is possible to regulate the force applied to the
workpiece by controlling the position of the end-effector rela-
tive to the workpiece. However, if the stiffness of the link is
high, any uncertadnty in the position of the workpiece, or er-
mrs in the position servo of the manipulator willinduce very
largecuttingforces.Eventually theseuncontrolledlargecutting
forceswillshorten the lifeof the grinding wheel and the ma-
nipulator.Also,the high stiffness(K/.)causes a high frequency
oscillationorunstable chatteringdue toreaction forcesfrom the
workpiece. This behavior can be easilyexplained by a root lo-
cns with increasingKL assuming that the position-controlled
robot is a linear mass-damper-spring system. Therefore, the
compliance of the manipulator becomes one of the important
parameters, and more compliance is desirablein the cutting
process with pure positioncontrol.
In thispaper, the compliance of a manipulator isderived
analytically,and itsmagnitude isrepresentedas a compliance
ellipsoid It isshown that the compliance willbe differentde-
pending on feedback gains,link flexibility,configurations,a_)d
directionof interest.Then, using a two link flexiblemanipula-
tor with an abrasivecut-offsaw, the experimental investigation
shows that the magnitude of the chatteringvarieswith the saw
cuttingangle due to the differentcompliance. The finalresults
show a range of cutting angles with acceptable behavior for a
point inthe workspace with a near circularcompliance ellipsoid
• Figure I.Block Diagram ofCutting Process
with a Position-ControlledManipulator
https://ntrs.nasa.gov/search.jsp?R=19910021215 2020-03-19T16:59:04+00:00Z
ww
w
1.2 Chatter
In the utilization of metal-cutting machine tools, vibra-
tions are oftenencountered. The contact between the toolsand
w_rkpiece givesriseto excessivevariationsof the cutting force
which endanger the lifeof the tool. These vibrationsbelong
to the classof self-excitedvibrations.The source of the self-
excitingenergy isin the cuttingprocess. Furthermore, in many
cases,the self-excitedvibrationsare mixed with forced vibra-
tionsexcitedby varioussources such as continuous spinning of
tiletools[3].In thispaper, the self-excitedand forcedvibrations
are referredto as chatter.
Considerable knowledge about the influenceofkinematical
parameters on the chatterhas been assembled. As yet,however,
neithera complete theoreticaldescriptionand analysishas been
accomplished nor reliablew_ys found foreliminatingchatcer in
grinding [3,4].By experimental trialand error,general guide-
lineshave been establishedto reduce the tendency for chatter.
Among theseare the use of soR-g_ade wheels,frequent dressing
of the wheel, changes in dressing techniques,reduction in ma-
terialremoval rttteand more rigddsupport of the workpiece [8].
Even though many parameters influencethe chattering,thispa-
per examines mainly the relatiotmhipbetween the cutting angle
m_l the compliance of the arm
3. DYNAMIC MODEL
3.1 Cutting Proce_
Exact modeling of a cutting process can be very compli-
cated I3,4,5,61. For simplicity, this paper assumes that the cut-
ring forces consist of the normal cutting force which is in the
direction of the approach angle of the saw and the tangential
cutting force wkich is perpendicular to the approach angle. The
relationship between the normal cutting force (F_) and tangen-
tial cutting force (Ft) can be assumed to be Coulomb friction
[71.i.e.
F,
_ -- _-_ ffi 0.3 _ 0.4
Notice that F, islargerthan Ft. Also,Fi and F, are a function
of the depth of the cut.
3.2 Flexible Manlpulator
Modeling a multiple link flexiblemanipulator isa compli-
cated procedure, The deflectionof the arm isapproximated as
WorkOlece
MOtlOn of TID
AI_IVe
Ft _
Figure 2. Definition of Cutting Angle
a finite series of separable functions which are the products of
mode shape functions O_(x) and time dependent generalized
coordinates ql,j ( t):
m
2=I
where i representsthe link number and j representsthe mode
number . The equation of the flexiblearm motion can be de-
rivedfrom severaltechniques,but the Lagrange's formulatloi_is
known for itssimplicityand systematic approach [9,10].Using
Jacobians to compute the velocityof a point,the kinematic and
potentialenergies are obtained by integratingthe velocityand
positionof the point over the totalsystem. These energies are
used in Lagrange's equations Therefore, the equation of the
motion is
M,_ M,/ "
-[6],+r":t," (1)
where q, contains the generalized rigid joint coordinates and q/
contains the generalized flexiblemode coordinates. 31v isthe
inertiamatrix and Kl. representsthe linkflexibi]ities. N, and
PC/include nonlinear terms such as the Coriolisand centrifugal
forcein each coordinate. Gr isthe gravity force. T represents
jointtorques and F representsan end point external forcewhen
the contact with environments occurs. Finally,d, and J/ are
the partitionsof the Jacobian matrix fora flexiblearm.
4. COMPLIANCE OF ARM
The equation of the dynamic motion for a flexible arm is
obtained in equation (1). Since the tip of the manipulator pro-
ceeds very slowly and in a small range during cutting, the mo-
tion can be assumed to be quasi-static and linear. Therefore,
the equation of the motion is simplified to the following form
assuming the acceleration terms and the velocity terms are neg-
ligible.
o .,, <?>
As shown above inthe above equation (2),the gravityforce
alsocontributestoa stiffnessforce.IfajointanglePD controller
isapplied to the flexiblearm, the jointtorque {"willbe
{r = - KpAq, - K_A4,
where Kp and K_ are the feedback gains. This input torque can
be interpreted as a spring and a damper force. Again q, can be
neglected in quasi-static motion. If we combine all these forces
for the stiffness matrix,
ww
w
v
0 KL \_ql] = LJl J
This stiffnessmatrix isalways invertible.Therefore, it can be
rewritten as.
,aq// 0 KL F (3)
Since this matrix shows the relationships between the end-point
external force and the joint coordinates, it is necessary to change
the joint space to the Cartesian space. Using the Jacobian re-
lationship which is
Substituting equation (3) into
[J, hl [_"
t
equation (4) willgive
+K,_ 0 ]-' [s,_] Fo XL LJiJ (_)
Since compliance isdefined as 'displacementper input force',
we may say
o l.Ji J (6)
This matrix, C, iscalledthe compliance matrix for a flexi-
ble arm with a jointangle PD controller.This compliance ma-
trixincludesnot only linkfiexibilities,but alsofeedback gains,
stiffnessand configurationsof the arm. The linkflexibilitiesare
represented by K/, and the feedback gains by Kp. The config-
urations of the arm are represented by J, and Gr. However,
the compliance matrix does not incorporate the mass proper-
tiesof the arm which alsomay influencethe arm's behavior. If
equation (6) isexpanded, itcan be rewrittenas
j_(OG, )-'Jf + J/KL-'J_C = k Oq, + Kp (7)
This form of the compliance matrix shows a major difference
between a flexiblearm and a rigidarm. In a rigidarm case,
KL isassumed to be very large.Therefore,we may ignore the
second term of equation (7) although itis the dominant term
in a flexiblearm case.
Since the compliance is represented with a matrix for a
multiple linkmanipulator, variousinput forcedirectionscause
differentdirectionsand sizesof displacements. This may be
explained with a linearalgebra concept. From equation (5),
we see that the compliance issimply a lineartransformation
that maps the end-point forceF in R 3 into a Cartesian space
displacement in R 3. The unit sphere in R a defned by
FTF < I
is a mapping into an ellipsoid in R _ defined by
= L,_(
(a) Unit Input Force Sphere Co) Compliance Ellipsoid
Figure 3. Compliance Ellipsoid
Axr(ccr)-tAX < 1
This ellipsoid has principal axes Alet, Aae2, A3e3 where el J.s a
unit vector and At is an e/genvalue of (CCP). We call this the
'compliance ellipsoid'. Therefore, a unit force F in the direction
a induces a displacement in the direction 3 as shown in Figure
3.
5. A CASE STUDY
A large experimental arm designated RALF (Robotic Arm,
Large and Flexible) has been constructed and is under computer
control. RALF has two degrees of freedom in the vertical plane.
The length of each link is about 10 feet. At the tip of RALF,
an abrasive cut-offsaw ismounted as shown inFigure 4. Using
the compliance ellipsoid,we explore the desirablecuttingangles
for RALF with acceptable chatteringbehavior.
5.1 Analysis
Based on the actualdynamic parameters of RALF, the dy-
namic equation isderived in the form of equation (I). Then_
actuator dynamics are assumed to be constant gainssincetheir
bandwidth is very high compared to the arm dynamics. The
actuator gains are included in the feedback gains Kp. The
nominal configurationduring cuttingisthe following:the first
jointangle is106.6 degrees,and the second jointangle is101.8
degrees. The compliance matrix iscomputed and itsmagnitude
isrepresented in R 2 with an ellipsoid.Simulation resultsshow
that the manipulator's axis of leastcompliance isat an angle
30 degrees with the horizontaland the axisof greatestcompli-
ance isat an angle of 120 degrees in Figure 5(a). Therefore,
the 120 degrees cutting angle isdesirableto produce the least
chatteringdue to itsgreatercompliance. Differentshapes ofthe
compliance ellipsoidcan be obtained at differentconfigurations.
For example, ifthe firstjointisat ii0 degrees and the second
jointisat 50 degrees,the compliance ellipsoidcan be shown as
in Figure 5(b).
B.2 Experiments
To measure the chattering in plane motion, two accelerom-
eters are mounted at the tip of RALF. One accelerometer mea-
sures the X direction vibration and the other accelerometer
measures the Y direction vibration referenced to the manipu-
lator base coordinates. Experiments use three different cutting
angles 0, 40 and 90 degrees. The cutting angles are shown as in
Figure 6 and the manipulator follows the given trajectory. Each
w
=_
v
w
Figure 4. RALF with an Abrasive Cut-off Saw
case assumes that it has the same cutting parameters except for
the cutting angle. For instance, cutting velocities are the same
for each case, and the same feedback gains are used too. The
workpiece is a half inch diameter steel bar and is much sti_er
than the manipulator system itself.
First, the abrasive cut-off saw moves very close to the work-
piece. Then, the saw is turned on without contact with the
workpiece. The vibrational signal is measured by a signal an-
alyzer, and its power spectrum is plotted in Figure 7(a) . The
firstnatural frequency isobserved at 4.5 Hz compared to 5 Hz
from the mathematical model. Also,another peak isobserved
at about 62 Hz . This frequency isbelieved to originatefrom
dynamic imbalance ofthesaw motor turning at 3800 rpm (63.3
I-Iz)by the manufacturer's data.
Second, the cut-offsaw followed the 0 degree desired tra-
jectoryby a jointanglePD control.The trajectoryiscomputed
based on Dickerson and Oosting'swork [11].It takes about 10
sec for the saw to go through the workpiece. The acceleration
power spectra are measured for 2 sec four times during cut-
tingand are averaged to eliminatethe influenceof non-periodic
noise. The same procedures are used foreach experiment. In
Figure 7(b), first peak is measured at 9 Hz. We may interpret
this shift of the first frequency due to the change in the bound-
a) Xh)
1 _ •
.li
(b) ,,_,
-2 -I 0 1 Z
J
!
"J I
/ /
, i
T
J
t
-I -I 0 I Z 3
('_/,_;,m'
Figure 5. Compliance Ellipsoid for RALF
(a) when 91 = 106.8 deg and _ = 101.8 deg
(b) when 8: = 110 deg and _ = 50 deg
ary condition when the saw touches the workpiece. Also, we
may notice that the rotation speed of the wheel isreduced duc
to the contact fTictionforce.
Third, when the saw cuts the workpiece at 40 degrees,the
firstnatural frequency(9 Hz) no longerdominates as before and
is mixed with other frequency signals in Figure 7(c). However,
the higher mode at 37 Hz becomes more noticeable. In othet
words, chattering becomes faster.
Fourth, the 90 degree cutting shows smaller magnitudes of
vibration in the Y direction, and the mode at 22 Hz becomes
important (See Figure 7(d)). We expect that this angle wil] give
the least chattering based on analytical analysis. Experimental
WorkDlect
Figure 6. Various Cutting Angle
(a) 0deg (b) 40deg (c) 90deg
ORIGINAL PAGE IS
OF POOR QUALITY
V(a)
Ydir
X dir
• .., , .,.
:-, :.
1
i
.... . ! 11
Co)
Xd_
x.lll _i
wow _ S_ECI 4^_g oxo Ip
- i i t.= _o0
w=d ,e _00_ 9 MZ
I [o 100
Figure 7. Power Spectrum Measurement in X-Y direction
(a) Without Any Contact with Workpiece
(b) 0 degreesCutting Angle
data fails to show a distinct advantage.
Finally, various cuts have been performed by a tele-operated
joystick under human control. Most of the cutting processes are
successfully accomplished without any severe chattering. How-
ever, its measurement is not included in this paper due to space.
Experimentally, the contact velocity is one of critical factors
which initiates chattering.
6. CONCLUSIONS
Taking advantage of the passive compliance of the flexible
manipulator, certain applications such as cutting a workpiece
are performed with pure position control. This provides a sim-
ple, inexpensive solution for certain applications that otherwise
could not be achieved with position control alone.
Both computer-controlled cutting and human-operated cut-
ting were performed with minor chattering. However, contact
velocityshould remain very small to reduce chattering.
The contact with the workpiece causes a shiftof the first
natural frequency of a flexiblearm due to the change of the
boundary conditions.Differentcuttinganglesproduce different
frequenciesof vibrationsdue to the differentcompliances inthe
-go _-
' 'Y dir
(d) " ;o ..
o-iY dbr A _
:7" = _ _,_ _'_
Figure 7. Power Spectrum Measurement in X-Y direction
(c) 40 degrees Cutting Angle
(d) 90 degrees Cutting Angle
dkection of forcing.
Analytical studies predict 120 degrees cutting as the most
desirable. However, this experimental investigation could not
show distinct differences in the magnitude of chattering," al-
though we may say that the 90 degree cutting angle generates
a smaller magnitude of chattering in the Y direction. The com-
pliance ellipsoid for our test bed is not elongated enough to
make distinct differences in compliance• A different configura-
tion could have made a more elongated ellipsoid, but further
experimental tests have not been conducted due to physically
limited location of the workpiece.
Acknowledgment
This work was partially supported by NASA Grant NAG
1-623. Also, we would Like to thank PCB PIEZOTRONICS,INC
for their generous donation of accelerometers.
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Experimental investigations of the effects of cutting angle on chattering of a flexible manipulator

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