Method of for grinding

There is disclosed a method of and apparatus for improving surface finishing and accuracy in a grinding operation employing a manipulator such as a robot which is coupled to a rotary grinding tool. The method comprises reducing the coupling effects between the tool motion whick is tangential to a workpiece surface and tool motion which is normal to the surface to obtain the optimum performance. The method involves adding compliance to the grinding tool in the tangential direction while maintaining higher stiffness in the normal direction. A robot is disclosed which is made in accordance with the method.

FIELD OF THE INVENTION 
This invention relates to surface grinding in general and more particularly 
to compliant manipulators, such as robots, and compliant tool or workpiece 
mounts and a method of designing them to perform the task of grinding with 
optimal performance. 
BACKGROUND OF THE INVENTION 
Most grinding operations, such as weld seam grinding and deburring of large 
castings, require a large workspace and dexterity with many degrees of 
freedom. Industrial robots appear to be suitable for these tasks, however, 
present day robots and manipulators in general have several technical 
problems that have prevented their successful application in the process 
of grinding and like processes. As a result, success in automating these 
tasks has been limited and many grinding applications have remained highly 
labor intensive in industry despite low productivity, high costs and 
hazardous working environments. 
In grinding applications, the robot manipulator is required to locate and 
hold the grinding tool in the face of large, vibratory forces which are 
inherent in the grinding process. Exposure to these unpredictable loads 
generally results in large deflections at the tip of the robot arm. These 
deflections degrade the process accuracy and the surface finish. In 
addition, the large vibratory loads may cause damage to the robot's 
mechanical structure. 
In conventional machine tools, large deflections are eliminated by 
designing for maximum stiffness in the whole structure. Unfortunately, it 
is not feasible for robot manipulators to have such high stiffness. For 
many robot applications including surface grinding the demands for wide 
workspace, dexterity and mobility with many degrees of freedom introduce 
kinematic constraints which make robots unavoidably poor in structural 
stiffness compared to conventional machine tools. 
The technical literature is replete with proposed solutions: 
As an alternative to high stiffness design, active feedback control has 
been applied to grinding robots for reducing dynamic deflections. One 
active control idea was proposed in which actuators are commanded to 
increase torques in the opposite direction to the deflections. This method 
reduces dynamic deflections in a certain frequency range. Generally, it is 
difficult for this control method to perform well over a wide frequency 
band because it must drive the entire, massive robot arm. 
Actively controlling wrist joints or local actuators which are located near 
the tip of the robot arm is easier and more effective than moving the 
whole arm, because the inertial forces are smaller. An active isolator has 
been applied to a chipping robot, where the isolator attached to the arm 
tip reduced the vibrations seen by the robot. A multi-axis local actuator 
was developed which compensates for positioning errors at the end point, 
in a limited range. 
For certain applications the stiffness of the robot can be significantly 
increased by directly contacting the workpiece. Tool support mechanisms 
have been developed which couple the arm tip to the workpiece surface and 
bear large vibratory loads. These mechanisms allow the robot to compensate 
for the tolerancing errors of the workpiece, as well as to increase the 
stiffness with which the tool is held. A local support mechanism has been 
applied to a drilling robot for part referenced positioning. 
Thus, a number of methods for improving performance and positioning 
accuracy have been developed, which can be used for a variety of machining 
applications. A key to successful application, however, is a sound 
understanding of the machining process, specifically the dynamic 
interactions between the tool and the robot manipulator. The grinding 
process, in particular, is a complicated dynamic process in which 
nonlinear and coupled dynamic behavior has a direct effect on the surface 
finish and accuracy. 
SUMMARY OF THE INVENTION 
In accordance with this invention, the relationship between tool vibration 
and the stiffness with which the tool is held is considered and the 
optimal tool suspension system is determined. A simple and effective 
solution to the robot grinding problem results which significantly reduces 
vibrations without additional actuators or active control. 
An optimal suspension system for compliant manipulators, such as robots, 
was determined through process analysis, simulation, and experimentation. 
Determination of the optimal suspension design was based upon an 
evaluation of the coupling between the motion of the grinding wheel in the 
normal and tangential directions relative to the surface of the workpiece 
and was formulated in terms of tool suspension stiffness. It was 
determined that strong coupling caused undesirably large vibrations and 
generally results in large low frequency waves and chatter marks on the 
workpiece surface. It was also determined that reducing the coupling 
between the wheel motions in the normal and tangential directions 
significantly improved the over all grinding performance. Low coupling was 
achieved when the stiffness in the direction normal to the desired 
workpiece surface was much larger than the stiffness in the direction 
tangent to this surface. Futhermore, with the optimal suspension design, 
the vibratory behavior during grinding was less erratic, the occurrence of 
low frequency waves in the workpiece surface were significantly reduced 
and both the accuracy and surface finish were improved. 
It was also determined that optimum results could be achieved by 
maintaining the direction of maximum stiffness as close as possible to the 
normal direction while maintaining at least 5 times higher compliance 
perpendicular to this direction. 
The above and other features of the invention will now be more particularly 
described with reference to the accompanying drawings and pointed out in 
the claims. It will be understood that the particular method expressing 
the invention is described by way of illustration only and not as a 
limitation of the invention. The principles and features of this invention 
may be employed in varied and numerous embodiments without departing from 
the scope of the invention, as for example, other machining processes such 
as rotary filing, high speed milling and internal gringing. Furthermore, 
the invention is not restricted to two dimensional applications.

BEST MODE OF CARRYING OUT THE INVENTION 
A model of a tool suspension system and the grinding process geometry 
hereinafter to be explained is shown in FIG. 1. The grinding wheel is a 
hard 6-inch diameter flat, cylindrical disk with a diameter much larger 
than the desired depth of cut, s.sub.o. The wheel is driven by a 2.5 
horsepower motor (not shown) rotating at 120 r.p.m. The depth of the cut, 
s.sub.o, is 20.times.10.sup.-3 inch and the feed rate is 0.33 inches per 
second along a 2-inch long, preground mild steel workpiece. 
The X axis, in FIG. 1 is directed tangent to the desired workpiece surface, 
while the Y axis is normal to this surface. The origin O is coincident 
with the center of the wheel, when the wheel is not deflected from its 
desired position. As the grinding process proceeds, the O-xy coordinate 
frame moves at the desired feed rate, v.sub.o. The deflections of the 
wheel are denoted in the formulae hereinafter presented by x(t) and y(t) 
(not shown in the Figures) and they are defined, with reference to the 
O-xy reference frame, to be positive when directed away from the 
workpiece. The variations of the wheel velocity in the x and y directions 
from the desired velocities are represented in the formulae x and y, 
respectively. 
The grinding wheel and tool are held by a robot manipulator shown in FIG. 
14. The tool suspension system shown in FIG. 1 represents the resultant 
characteristics of a robot arm and an end-effector which couples the 
grinding tool to the tip of the robot arm. 
In general, stiffness matrices of multi-axis mechanical systems have 
principal axes along which the stiffness is decoupled and can be 
represented by individual springs. Although the principal axes for damping 
are generally not coincident with those for stiffness, this will be 
assumed here for simplicity. Thus, the resultant characteristics can be 
represented in FIG. 1 by two springs and two dampers k.sub.p, k.sub.q and 
b.sub.p, b.sub.q, directed along the principal directions p and q 
accordingly. The p and q directions are orthogonal and the origin of the 
O-pq coordinate frame is coincident with that of the O-xy frame. The 
coordinates, p(t) and q(t) (not shown in the Figures), represent the 
deflections of the grinding wheel from its desired position in the p and q 
directions. The angle .alpha. (FIG. 1) represents the rotation of the O-pq 
coordinate frame relative to the O-xy frame and will be referred to 
hereinafter as the "structural stiffness orientation". The spring 
constants k.sub.p, k.sub.q and the structural stiffness orientation 
.alpha. are the primary tool suspension design parameters. These 
parameters are optimized through their effect upon the grinding 
performance. 
First, the dynamic behavior of the tool suspension system was considered. 
Let m be the mass of the grinding tool or wheel, then the equations of 
motion of the tool suspension system in the principal directions are given 
by 
EQU mp+b.sub.p p+k.sub.p p=f.sub.q 
EQU mq+b.sub.q q+k.sub.q q=f.sub.q (1) 
where f.sub.p and f.sub.q are the components of the forces acting on the 
grinding wheel in the p and q directions respectively. 
The reaction force, F, acts upon the wheel during grinding. This reaction 
force has components in both the normal, y, and tangential, x, directions 
which are represented by F.sub.n and F.sub.t, respectively. The 
relationship between these components is given by 
EQU F.sub.t =.mu.F.sub.n, .mu.=tan .theta. (2) 
where .mu. is similar to a friction coefficient. The coefficient .mu. is 
assumed constant for simplification. This is equivalent to assuming a 
constant force angle, .theta.. This approximation is only made in the 
simulations and in no way limits the applications of this invention. 
Surface machining processes, such as grinding, are characterized by strong 
coupling between the normal and tangential directions, which results from 
the inherent coupling in the reaction force, F. As the wheel moves in the 
tangential direction, the tangential reaction force, F.sub.t, varies. At 
the same time, the normal reaction force, F.sub.n, varies according to 
equation (2), which affects the wheel motion in the normal direction. This 
coupling of the wheel motion in the x and y directions was analyzed and 
the dependence of the vibratory behavior of the grinding wheel on this 
coupling will be shown. 
The components of the reaction force in the principal directions, f.sub.p 
and f.sub.q, are given by 
EQU f.sub.p =Fsin (.alpha.+.theta.) 
EQU f.sub.q =Fcos (.alpha.+.theta.) (3) 
Substituting equation (3) into equation (1), eliminating the scalar force 
of magnitude F and taking the laplace transform yields the following 
relation between the behavior along the two principal directions. 
##EQU1## 
The final dimensions of the workpiece are directly determined by the 
behavior of the grinding wheel in the y direction. Deflections in the x 
direction, on the other hand, have no direct effect on these dimensions. 
Thus, the behavior of the wheel in the y direction is of primary concern. 
The following coordinate transformations are now introduced. 
EQU x=p cos .alpha.-q sin .alpha. 
EQU y=p sin .alpha.+q cos .alpha. (5) 
The resulting equation is 
##EQU2## 
where B.sub.t, K.sub.t and B.sub.n, K.sub.n are each a different function 
of .alpha., k.sub.p, b.sub.p, k.sub.q, and b.sub.q and represent the 
damping and stiffness properties of the suspension system in the normal 
and tangential directions. 
The transfer function G(s) in equation (6) represents the effect of 
behavior in the x direction on behavior in the y direction. This accounts 
for the coupling caused by the relation described in equation (2). It has 
been stated that vibrations are more likely in the tangential direction 
than the normal direction because of relatively low process stiffness in 
that direction, and vibrations in the tangential, x, direction cause 
pulsating normal forces, F.sub.n, which cause deflections of the wheel in 
the normal, y, direction. From this observation, it follows that the 
coupling must be reduced as much as possible so that the motion in the 
normal direction, which directly determines the final workpiece 
dimensions, is not significantly disturbed by large vibrations in the 
tangential direction. 
If only high frequency vibrations are considered the ms.sup.2 terms in the 
transfer function dominate and G(s) reduces to cot .theta.. However, it 
has been shown that high frequency vibrations do not produce waves on the 
workpiece surface. In other words, for a given feed rate, vibrations above 
a certain frequency will not directly effect the final workpiece 
dimensions. 
Low frequency vibrations in the normal direction will show up directly as 
waves on the workpiece surface. For low frequency vibrations the stiffness 
terms dominate in equation (6). Then the transfer function G(s) reduces to 
##EQU3## 
and .kappa. represents the static coupling between the deflections in the 
normal and tangential directions. Since the design parameters .alpha., 
k.sub.q, and k.sub.p are involved in .kappa., they effect the coupling for 
low frequency vibrations. The effect of .alpha. and d on .kappa. is shown 
in FIG. 2 for force angles of 45.degree. and 30.degree.. When the value of 
.kappa. is near 1, the deflections in the normal and tangential directions 
are approximately equal and the behaviors in the two directions are 
strongly coupled. This strong coupling occurs when the p or the q axis is 
directly aligned with the direction of the grinding force, .theta. or when 
the two spring constants, k.sub.p and k.sub.q, are approximately equal. 
The value of .kappa. is small when the structural stiffness orientation, 
.alpha., is close to O and d is large, or when .alpha. is close to 
90.degree. and d is small. Both sets of conditions describe the same 
configuration for the springs, which requires K.sub.t &lt;&lt;K.sub.n. Also the 
value of .kappa. is large when .alpha.=O and d is small, or when 
.alpha.=90.degree. and d is large, namely when K.sub.t &gt;&gt;.sub.n. In both 
cases, .kappa.&lt;&lt;1 and .kappa.&gt;&gt;1, the two directions are weakly coupled. 
Vibrations during the grinding process become large when the two axes are 
strongly coupled, .kappa. near 1, and the vibrations are significantly 
reduced for suspension designs which result in weak coupling, 1/2 much 
different from 1. 
Grinding Force Analysis 
In this section, a macroscopic representation of the grinding force was 
derived from basic empirical relationships. The goal was not to model the 
process at a microscopic level but to develop an aggregate representation 
of the grinding force, which produces behavior similar to that observed in 
practice and experimentation, when introduced into the original model, 
shown in FIG. 1. 
The grinding force formulation was based on two fundamental empirical 
relations. The first was presented in equation (2). The second equation is 
EQU Z.sub.w =.LAMBDA..sub.w F.sub.n 
EQU F.sub.n =CZ, C=1/.LAMBDA..sub.w (9) 
In the first equation above, the variable Z.sub.w represents the volume 
removed from the workpiece surface per unit time. .LAMBDA..sub.w is the 
"Metal Removal Parameter". Since, only the workpiece material removal was 
considered, Z.sub.w is replaced by Z and the equation was rewritten as in 
the second equation (9). 
In conventional grinding machines, the structure stiffness is high in every 
direction and it can be assumed that x, y, x, and y are zero. For these 
conditions, the volume removal rate, Z, is equal to v.sub.o s.sub.o. 
However, when relatively compliant robots are exposed to large 
unpredictable forces at the tip of the arm, large deflections result and 
the conventional formulation is not sufficient. In fact, even for 
conventional grinding, when chatter occurs, the deflections and velocity 
variations are not zero. It has been stated that the feed velocity varies 
between (v.sub.o -x.sub.min) and (v.sub.o +X.sub.max) in grinding and that 
this phenomenon is similar to type B chatter in lathes. Consequently, the 
velocity v.sub.o in the equation for Z must be adjusted accordingly. 
Furthermore, it has been found that the actual depth of cut is not equal 
to the desired depth of cut when there are deflections of the grinding 
wheel in the normal direction and s.sub.o must be replaced by (s.sub.o -- 
y). Thus, the actual volume removal rate, which results from wheel motion 
in the tangential direction, is 
EQU Z.sub.t =(v.sub.o -x) (s.sub.o -y) (10) 
As seen in equation (9), the normal grinding force is directly proportional 
to the volume removal rate. Thus, the normal grinding force, 
F.sub.n.sup.x, resulting from motion of the grinding wheel in the 
tangential, x, direction is given by 
EQU F.sub.n.sup.x =C.sub.t (v.sub.o -x) (s.sub.o -y) (11) 
In addition, if the wheel is moving in the negative y direction, material 
is removed in the normal direction as in plunge grinding. The volume 
removal rate, which results from the motion of the grinding wheel in the 
normal direction only, is approximately proportional to y and the actual 
depth of cut, (s.sub.o -y), and is given by 
EQU Z.sub.n =-y.lambda.(s.sub.o -y) (12) 
where .lambda. is a constant derived from the process geometry, and the 
normal grinding force, F.sub.n.sup.y, resulting from motion in this 
direction is given by 
EQU F.sub.n.sup.y =-C.sub.n y(s.sub.o -y) (13) 
where C.sub.n =.lambda.C 
It was found in the simulations that, when the vibratory behavior during 
grinding is erratic and the grinding wheel is bouncing on the workpiece 
surface, the volume removal rate in the normal direction produces large 
forces and Z.sub.n can not be neglected. However, during relatively stable 
grinding conditions, the velocity of the grinding wheel in the normal 
direction, y, is much smaller than its velocity in the tangential 
direction, (v.sub.o -x). Thus, under these conditions the grinding force 
resulting from motion of the wheel in the tangential, x, direction 
dominates. Consequently, to account for both stable and erratic grinding 
conditions, the normal and the tangential volume removal terms are both 
required. 
The total grinding force in the normal direction can be approximated by 
adding equations (11) and (13). The resulting equation is 
EQU F.sub.n =C.sub.t (v.sub.o -x) (s.sub.o -y)-C.sub.n (s.sub.o -y)y (14) 
where C.sub.t =C from equation (9) and C.sub.n =.lambda.C. 
In adding these two effects, we have neglected terms of the form xy and any 
higher order terms which result from an exact solution for the volume 
removal rate. However, when the actual depth of cut is much smaller than 
the radius of the grinding tool, which is generally the case, these terms 
can be neglected. As will be shown hereinafter, this approximation for the 
normal grinding force, along with the relation between the normal and 
tangential grinding forces, equation (2), provides good agreement with the 
behavior observed in experimentation when introduced into the grinding 
process model, FIG. 1. 
Before the equations of motion are simulated, certain additional 
nonlinearities must be introduced. For example, if y&gt;s.sub.o the wheel is 
no longer in contact with the workpiece and C.sub.t =C.sub.n =0. In 
addition, if y&gt;0 the wheel is moving away from the workpiece, in the 
normal direction, and although material can still be removed as a result 
of motion in the tangential direction, no material is removed in the 
normal direction. Therefore, under these conditions C.sub.n =0 and 
material is only removed in the tangential, x, direction with an actual 
depth of cut equal to (s.sub.o -y). This is valid as long as the actual 
depth of cut is much smaller than the radius of the cylindrical grinding 
wheel. Similarly, if x&gt;v.sub.o no material is removed as a result of wheel 
motion in the x direction and C.sub.tl= 0. 
Thus, the resulting grinding force representation and, consequently, the 
equations of motion are highly nonlinear. It will be shown hereinafter 
that the resulting equations of motion can generate nonlinear behavior 
similar to that observed in the experiments. 
Simulation 
The equations developed in the previous section were then simulated and the 
conclusions of the simulations were verified experimentally. The objective 
of the simulations and the experiments was to determine the optimal 
combination of directional stiffness properties and structural stiffness 
orientation. 
Experiments were run with a 6-inch diameter, 1 inch thick, hard cylindrical 
grinding wheel, a 2.5 hp grinding tool and a 2-inch long preground mild 
steel workpiece. Experiments were run for a wide range of wheel speed, 
desired depth of cut and feed rate. In the specific experiments these 
parameters were 1200 rmp, 20.times.10.sup.-3 inch and 0.33 inch/sec 
respectively. 
A typical industrial robot has endpoint compliance properties which vary 
with arm configuration and direction. To emulate this condition in the 
grinding experiments, a compliant wrist was designed to permit the 
variation of the tool suspension stiffness properties. The wrist was also 
equipped with strain gages for force measurement, and the experimental 
data was sampled at 500 Hz by a Digital PDP/11 computer. 
First, the validity of the model was verified by comparing the simulated 
output with the behavior observed in experiments. The simulated wave form 
was generated by varying the parameters involved in equation (14) to find 
the best match with the experimental data. This parameter matching was 
repeated for several sets of data corresponding to different values of the 
design parameters, k.sub.p, k.sub.q and .alpha. and the process parameters 
v.sub.o and s.sub.o until satisfactory agreement was obtained for an 
number of different grinding conditions and directional stiffness 
properties. 
The experimental data and the simulation for a typical set of grinding 
conditions are shown in FIG. 3. FIG. 3-(a) shows the results of an 
experiment in which .alpha.=-45.degree. and k.sub.q =10k.sub.p. The higher 
frequency vibration in the experimental data was about 20 Hz, which was 
the same as that of the wheel rotation speed. Thus, this vibration is a 
forced vibration resulting from wheel imbalance. In the simulation, a 
forcing term which represents the wheel imbalance was introduced. The 
results of the simulation are shown in FIG. 3-(b) for the same directional 
stiffness properties and structural stiffness orientation as in the 
experiment. In each case, a similar wave form was observed at roughly the 
same frequency and amplitude. Thus, with the identified model parameter 
values, the simulated response is in satisfactory agreement with the 
experimental behavior. 
The objective of the simulations was to investigate the relationship 
between the wheel vibrations and the tool suspension design parameters, 
particularly the static coupling parameter, .kappa.. The real wheel 
vibrations observed in the experiments were highly complex and were 
influenced by such disturbances as wheel imbalance, irregular wheel wear 
and burrs. These disturbances and parameter changes are difficult to model 
precisely. However, in order to effectively evaluate the tool suspension 
design, the response to deflections caused by such disturbances had to be 
evaluated. Thus, simulations were run to examine the performance of 
various tool suspension designs in the face of unanticipated wheel 
deflections which might result from such disturbances. 
Typical response to disturbances that deflect the wheel in the x and y 
directions are shown in FIG. 4. These disturbances were considered by 
changing the initial conditions in the simulations. The initial 
deflections are x.sub.o in the x direction and y.sub.o in the y direction. 
The vibratory behavior was evaluated in terms of two representative 
parameters which were defined for the simulated waveforms: the maximum 
peak amplitude, Y.sub.max, and the settling time, t.sub.s. As mentioned 
earlier, the final workpiece dimensions were directly determined by the 
dynamic behavior of the wheel in the y direction. The maximum peak 
amplitude, Y.sub.max, was used as a measure of the wheel vibration in the 
y direction in response to deflections in the x direction as shown in FIG. 
4-(a). For deflections in the y direction, the behavior is shown in FIG. 
4-(b) for the same conditions. In this case, the time t.sub.s, required 
for the vibration amplitude to reach y.sub.o /4 is used to characterize 
the vibration. 
The effects of the static coupling parameter, .kappa., on the grinding 
performance with .alpha. set to zero and .theta.=45.degree. is summarized 
in FIG. 5. Both the maximum peak amplitude, Y.sub.max, and the settling 
time, t.sub.s, are maximum when .kappa. is approximately 1. Therefore, the 
worst behavior was observed when .kappa. is equal to 1, which occurs when 
K.sub.n =K.sub.t. The best disturbance rejection was obtained when 
.kappa.&lt;&lt;1 or .kappa.&gt;&gt;1, which occurred when the stiffness in the X and Y 
directions are far apart. 
As seen in FIG. 5, grinding performance is directly effected by the static 
coupling parameter, .kappa., which is proportional to the stiffness ratio, 
K.sub.t /K.sub.n. The vibratory behavior is also effected by the grinding 
force parameters, C.sub.t and C.sub.n. FIG. 5 shows the vibratory 
characteristics associated with grinding force parameter ratios, C.sub.n 
/C.sub.t =1 , 2, 4 and 10. The simulations in which .kappa.=1 consistently 
correspond to the worst grinding conditions for each grinding force 
parameter ratio. However, as the ratio C.sub.n /C.sub.t increases, the 
poor behavior observed near .kappa.=1 is alleviated, as shown in FIG. 5. 
As C.sub.n /C.sub.t increases the effective impedance of the process in 
the normal direction becomes larger than that in the tangential direction. 
As a result, vibrations are more likely in the tangential direction 
because the process impedance in that direction is much smaller. Thus, for 
C.sub.n /C.sub.t large, the importance of avoiding .kappa.=1 is less 
significant. However, this is not the case in typical robot surface 
grinding applications, thus .kappa.=1 should be avoided. 
The grinding force parameters were identified by the following procedure. 
First, the normal grinding force resulting in the simulations under stable 
grinding conditions when, y&lt;&lt;(v.sub.o -x), was matched with that observed 
under the same conditions in the experiments by varying C.sub.t. Once 
C.sub.t was determined, the normal grinding force parameter, C.sub.n was 
found by matching the behavior of the simulations with those in the 
experiments for a number of grinding conditions for which terms involving 
the velocity of the wheel in the normal direction, y, could not be 
neglected. It was found that C.sub.n /C.sub.t =2 provides the best match 
with experimental data, and for this grinding force parameter ratio the 
static coupling parameter, .kappa., is very important in evaluating 
grinding performance, as seen in FIG. 5. 
From FIG. 5, it is clear that the condition when k.sub.q =k.sub.p must be 
avoided and k.sub.p &lt;&lt;k.sub.q is desirable. However, the optimal 
structural stiffness orientation, .alpha., still had to be determined. 
Without loss of generality, the optimal orientation angle can be found for 
k.sub.p &lt;&lt;k.sub.q. 
FIG. 6 shows the effect of the structural stiffness orientation, on this 
normalized performance measure Y.sub.max /x.sub.o. In this case, the force 
angle was set to 30.degree. to emulate conditions observed in experiments 
and to permit comparison. The worst behavior was then observed when 
.alpha. equal 45.degree. or -45.degree., which correspond to values of 
.kappa. close to 1. The best behavior occurred when .alpha. equals zero. 
From FIG. 5 it appears that either high or low values of .kappa. will 
provide good performance. High values of .kappa. occur when K.sub.n 
&lt;&lt;K.sub.t and low values of occur when K.sub.t &lt;&lt;k.sub.n. Thus, it 
remained necessary to determine which of these cases will provide the best 
grinding performance. Many investigators have introduced springs in the 
normal direction to improve stability. However, with high compliance in 
the normal direction it is difficult to accurately locate the grinding 
wheel in the face of unpredictable dynamic loads. In addition, from the 
complete equations of motion it was found that the steady state deflection 
in the y direction, Y.sub.ss, is directly related to the stiffness in the 
normal direction by 
##EQU4## 
Since Y.sub.ss decreases as K.sub.n increases, it is desirable to have 
K.sub.n as large as possible. This explains the difference between the 
grinding performance when .alpha. equals 0.degree. and 90.degree., shown 
in FIG. 6. Although .kappa. is much different from 1 in both cases, the 
normal stiffness when .alpha. is 90.degree. is low and consequently the 
steady state error is large, and large deflections occur in the y 
direction with this tool suspension design. Thus, the low value of the 
static coupling parameter, .kappa., which corresponds to .alpha.=0 and 
k.sub.p &lt;&lt;k.sub.q, provides the best overall grinding performance. 
Experimentation 
To verify the conclusions of the simulations, the optimal structural 
stiffness orientation was also determined experimentally. A schematic of 
the wheel and workpiece orientation for the experiments is shown in FIG. 
7. The grinding wheel was suspended by a rectangular beam, for which the 
directions of the principal axes were adjustable, and K.sub.p &lt;&lt;K.sub.p. 
The actual force data for these experiments is shown in FIGS. 8 through 11 
for .alpha.=-45.degree., 0, 45.degree. and 90.degree.. From the force data 
in FIGS. 8 and 10, it was clear that orientation angles of 45.degree. and 
-45.degree. produced highly erratic behavior during grinding. When 
.alpha.=-45.degree., the high stiffness direction was nearly aligned with 
the resultant grinding force and the behavior appeared to be a high 
frequency limit cycle with low frequency beats. This phenomenon is common 
in grinding practice and it was found that the number of beats observed in 
the force data showed up directly on the workpiece surface as an 
equivalent number of undesirable low frequency waves. When .alpha.= 
45.degree., the low stiffness direction is nearly aligned with the 
resultant grinding force and the deflections are large and erratic; this 
appears to be the worst tool suspension design. In FIGS. 9 and 11 it 
appears that orientation angles of 0.degree. and 90.degree. both resulted 
in relatively stable grinding. For each of these tool suspension designs, 
it was found that an excellent surface finish could be obtained. To 
determine which of these designs provides the best overall grinding 
performance, it was necessary to consider the magnitude of the deflections 
as well as the degree of stability in the grinding forces. 
The maximum deflections for each of the experiments are shown in FIG. 12 in 
the same form as the simulated data in FIG. 6 to permit comparison, except 
in the experiments the disturbances were unknown and the maximum 
deflection Y.sub.max was not normalized. In addition to the maximum 
deflection in the y direction, the average deflection is also plotted in 
FIG. 12 to provide a measure of the actual depth of cut. 
As found in the simulations, when .alpha.=90.degree. and k.sub.p &lt;&lt;k.sub.q 
the stiffness in the normal direction is low and consequently, the average 
deflection, Y.sub.avg, caused by the grinding force is relatively large. 
This results in very low material removal rates and very poor accuracy, 
since Y.sub.avg represents the average difference between the actual and 
desired depth of cut. Thus, although both .alpha.=0.degree. and 90.degree. 
provide stable grinding, .alpha.=0 provides the best overall grinding 
performance. 
Experiments were also run for .alpha.=0 and k.sub.p =k.sub.q. To emulate 
grinding with a robot manipulator, the stiffness of the wheel suspension 
beam in the two principal directions was chosen to be equal to the 
endpoint stiffness of a typical industrial robot arm. This is necessary 
because the maximum suspension stiffness with which the grinding tool is 
held is limited by the stiffness of the robot arm. With typical robot 
grinding conditions and K.sub.n =K.sub.t, which corresponds to .kappa.=1, 
the behavior shown in FIG. 13 was recorded. The behavior under these 
conditions was much more erratic than that observed for .alpha.=0 or 90 
and k.sub.p &lt;&lt;k.sub.q. This again confirmed the conclusion that .kappa.=1 
is not a good design for the grinding robot tool suspension system. 
The best overall behavior is observed in the experiments when .alpha.=0 and 
k.sub.p &lt;&lt;k.sub.q. Under these conditions K.sub.n &gt;&gt;K.sub.t and the 
average deflection and the maximum vibration amplitudes are both minimum, 
as shown in FIG. 12. Thus, for these conditions the accuracy, stability, 
and material removal rate are optimal. 
End-Effector Compliance Design 
It has been shown that the optimal tool suspension design occurs when 
K.sub.n &gt;&gt;K.sub.t. A general design procedure was developed which utilizes 
this conclusion to determine the optimal grinding end-effector design for 
an industrial robot. The goal is to achieve high stiffness in the 
direction normal to the desired workpiece surface and relatively low 
stiffness in the direction tangent to this surface. The tool suspension 
system is comprised of both the main robot arm and the end-effector, which 
couples the main arm to the grinding tool. 
The endpoint compliance of the main robot arm is determined by the 
structural stiffness of the arm linkage and the servo stiffness of the 
individual joint actuators. It is possible to modify the resultant 
endpoint compliance, C.sub.r, by introducing additional mechanical 
compliance at the tip of the robot arm. 
For simplicity, consider the two degree-of-freedom manipulator shown in 
FIG. 14. The O.sub.w -xy coordinate frame is defined as before so that the 
y axis is directed normal to the desired workpiece surface and the x axis 
is tangent to this surface. The resultant compliance matrix defined with 
respect to this workpiece coordinate frame is then given by 
EQU C.sub.r.sup.xy =C.sup.xy.sub.arm +C.sub.e.sup.xy (14-A) 
where C.sup.xy .sub.arm is the endpoint compliance matrix of the main robot 
arm defined with respect to the workpiece coordinate frame, O.sub.w -xy, 
and C.sub.e.sup.xy represents the additional compliance introduced at the 
endpoint and is also defined with respect to the workpiece coordinate 
frame, O.sub.w -xy. These two terms can be added to obtain the resultant 
compliance matrix, C.sub.r.sup.xy, since the additional compliance is in 
series with the robot arm compliance. In other words, if a force is 
applied to the grinding wheel, the robot arm and the end-effector will 
each see the same force. 
For a typical industrial robot, the robot arm compliance can be determined 
by direct measurement or by calculation from the structural stiffness of 
the arm linkage and the servo stiffness of the individual joint actuators. 
This endpoint compliance will generally vary with both arm configuration 
and the direction of the applied force. For the two degree-of-freedom 
manipulator shown in FIG. 14, the following notation will be introduced to 
represent the components of the robot arm compliance matrix when defined 
in the O.sub.w -xy coordinate frame. 
##EQU5## 
The O.sub.w -uv coordinate frame is then defined, as shown in FIG. 14 so 
that the end-effector compliance matrix defined with respect O.sub.w -uv 
is given by 
##EQU6## 
where k.sub.u and k.sub.v represent the stiffness of the end-effector 
along the corresponding principal directions, and .phi. represents the 
orientation of the O.sub.w -uv coordinate frame with respect to the 
O.sub.w -uv coordinate frame. To obtain the resultant compliance matrix, 
it is first necessary to transform C.sub.e.sup.uv into C.sub.e.sup.xy by 
EQU C.sub.e.sup.xy =R.sup.T C.sub.e.sup.uv R (17) 
where R is the 2.times.2 rotation matrix associated with the angle .phi.. 
As stated earlier, the goal was to achieve the condition, K.sub.n 
&gt;&gt;K.sub.t. This goal can be represented in the workpiece coordinate frame 
as an optimal compliance matrix, C.sup.xy.sub.opt. This optimal tool 
suspension compliance matrix is derived from the conclusions of the 
simulations and experimentation and is given by 
##EQU7## 
where C.sub.xx =1/K.sub.n and C.sub.yy =1/K.sub.t. 
An optimal set of design parameters .phi., k.sub.u and k.sub.v can now be 
determined by equating the resultant compliance C.sub.r.sup.xy with the 
optimal compliance C.sup.xy.sub.opt, given in equations (15) and (18) 
respectively. The primary end-effector design parameters are then 
determined by the following functions of the components of the robot arm 
compliance matrix and the optimal compliance matrix. 
##EQU8## 
The following design procedure must then be followed. First, the compliance 
matrix of the main arm is evaluated for a given workpiece location. The 
minimum compliance obtainable at the tip of the robot arm is limited by 
the minimum main arm compliance. Thus, the second design step is to chose 
C.sub.yy equal to this minimum value. Then C.sub.xx is chosen so that 
C.sub.xx &gt;&gt;C.sub.yy. Finally, the primary design parameters can be 
determined directly from equation (19). It is also possible to determine 
the optimal workpiece location and orientation by a similar procedure. 
As a specific example, 
Cyy.apprxeq.1/K.sub.min =C.sup.a.sub.max where K.sub.min is the minimum 
translational stiffness at the tip of the arm. For the robot employed in 
the experiments, K.sub.min .perspectiveto.1.times.10.sup.5 newtons per 
meter. 
Choose Cxx.gtoreq.5 Cyy, Cxx=1.times.10.sup.-4 M/N, for example, will 
eliminate coupling effects is this case. 
Substituting Cxx and Cyy into equation (19) and with the workpiece normal 
to the maximum robot endpoint stiffness, the values for .phi., k.sub.u and 
k.sub.v are 
EQU .phi.=0 
EQU K.sub.u .gtoreq.1.times.10.sup.5 newtons per meter 
EQU K.sub.v .multidot..ltoreq.1.times.10.sup.4 newtons per meter 
The optimal tool suspension design for grinding with robots was determined 
through dynamic analysis, simulation and experimentation. First, the 
grinding wheel suspension system was characterized by the coupling between 
the normal and tangential motions. Second, a nonlinear mathematical model 
of the grinding force was formulated and the model parameters were 
selected through comparison with experimental grinding force data. 
Utilizing this model, it was shown that the worst vibratory behavior 
results when the normal and tangential motion of the grinding wheel are 
strongly coupled. It was found that the best grinding performance was 
achieved when the stiffness of the tool suspension system in the direction 
normal to the desired workpiece surface is much larger than the stiffness 
in the direction tangent to this surface. This conclusion was verified 
through both simulation and experimentation. Finally, a design procedure 
for determining the optimal end-effector design for an industrial robot 
was developed. 
The proposed design procedure has been utilized to design an end-effector 
for a heavy duty grinding robot. The end-effector is comprised of and 
optimal tool suspension system and a 6-inch diameter cylindrical grinding 
wheel with a 2.5 hp motor. The design was tested on the grinding of weld 
seams and it was found that over 30 mils could be removed from the weld 
seam in one pass without noticeable chatter and with a good surface finish 
.