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Path generation for industrial robots
M. Nystrom1, M. Norrlof2
Department of Electrical Engineering,Linkoping University,
SE-581 83 Linkoping, Sweden
1. e-mail: [email protected]. e-mail: [email protected]
Abstract
Path generation in Cartesian space is discussed and theidea of
using lines and arcs to create general paths ispresented. To make
smooth paths in Cartesian spacezones are introduced and two
different ways to inter-polate zones are given. An algorithm for
transforminga Cartesian path into a cubic spline representation
injoint space is presented and also evaluated in an ex-ample. An
important aspect of the proposed algorithmis that the resulting
path in joint-space has a boundedCartesian space error.
1 Introduction
There is a growing interest in academia and indus-try for new
algorithms, methods and tools in gen-eration and optimization of
trajectories for indus-trial robots [4]. This development is
motivated bythe increasing need for improved motion perfor-mance in
several robot applications such as lasercutting, laser welding,
machining, assembly andmaterial handling.
The trajectory generation problem can be di-vided into two
separate problems. The first,which is covered here, deals with the
path plan-ning or path generation. This is a strictly geo-metrical
problem but since it serves as an inputto the next step, the
trajectory planning and opti-mization, it does have an impact on
the resultingtrajectory. The two problems are often consid-ered as
one, see for example [4, 1, 2, 6]. Many ofthe early contributions
only covers point-to-pointThis work was supported by the VINNOVA
Compe-
tence Center ISIS at Linkoping University
motion while in many real applications where in-dustrial robots
are used the complete path mustbe specified. In [3] the Cartesian
path is approx-imated by line segments. From line segments itis
easy to take the step to represent the path usinggeneral splines,
this is discussed in e.g. [7]. Tocontrol the manipulator it is
necessary to trans-form the Cartesian path into a joint space
repre-sentation. This step can also include the dynamicoptimization
as described in [3, 2].
In the first part of this contribution path gen-eration in
Cartesian space using lines and circlesegments is discussed. One
problem in Cartesianspace is how the lines and circles should be
con-nected to each other to create a smooth path. Thesolution is
based upon the introduction of zonesin which an interpolation
between the two lines,the two circles or the circle and the line
can bedone.
The second part of this contribution deals withthe generation of
paths in joint space. This in-cludes transforming the Cartesian
path to jointspace using the kinematic model [10, 8] of therobot
and also to interpolate a path in joint spaceusing splines. An
algorithm for spline interpo-lation is presented and ideas on how
to adjustthe length of the splines in order to keep a cer-tain
level of path accuracy in Cartesian space aregiven.
Finally an example is presented where the algo-rithms for path
generation are applied to a Carte-sian line-circle path. This paper
is based upon theresults presented in [8], [9], and [7].
1
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2 The path in Cartesian spaceThe first step in path generation
is to create a pathin Cartesian space. Here it will be assumed
thatthe robot has only 3 degrees of freedom (DOF)and therefore only
the position of the path is con-sidered.
2.1 Path primitives
The path followed by the robot end effector (ortool) is in
general composed by connecting anumber of lines and arcs, where
each line andarc is referred to as a segment. Each segment hasa
parametric representation p(lc) where lc is thelength of the
segment (in m). A line segment isrepresented by a first order
spline and the arc isrepresented by p(lc) = Tp(lc) where
p(lc) =(r cos(lc/r) r sin(lc/r) 0 1
)T(1)
and p(lc) is expressed in a local coordinate framewith origin in
the center of the circle. r is the ra-dius of the circle and lc is
the index varying fromzero to the length of the arc. T is a
homogeneoustransformation matrix (see e.g. [10]) expressingthe
coordinate transformation between the localcoordinate frame and the
global coordinate frame.
2.2 Introduction of zones
In order to make the path smooth, a circular zoneis introduced
around the intersection points of ev-ery two adjacent segments, as
shown in Figure 1.Within this zone, the path is allowed to
deviatefrom the segments as long as the path does notleave the
zone. The actual path is interpolatedfrom the two adjacent
segments.
zon 1
zon 2
segment 1segment 2
segment 3
Figure 1: Three segments, an arc and two lines.The two zones
used in the interpolation of asmooth path are shown as dashed
circles.
During the interpolation a normalized index isused. This means
that each segment is given anew index l, where l [0, 1]. In the
intersectionpoint with zone j, index for segment i, li, has
thevalue lizj where j = i or j = i 1.
The path inside the zone j uses an internal in-dex lzj , which
is based on the normalized indexof the second segment of the zone.
For zone 1 inFigure 1 this means that lz is based on the
normal-ized index for segment 2. In general for a zone j:lzj
[l(j+1)zj , l(j+1)zj ].
If the robot was programmed to stop at the con-nection point of
two adjacent segments, then thedistance from the beginning of the
zone to theconnection point could be seen as a braking dis-tance.
The distance from the connection point tothe end of the zone could
similarly be consideredas acceleration distance. If the braking and
the ac-celeration take place simultaneously during a time , then a
point pi,1 should be reached on the firstsegment and a point
p(i+1),1 on the second seg-ment. If these two points are seen as
vectors fromthe connection point of the two segments, the
cor-responding point on the interpolated path is givenby pi,1 +
p(i+1),1, see Figure 2. This procedurecontinues until, on the first
segment, the connec-tion point is reached, and on the second
segment,the end point of the zone is reached.
y0O 0
z0
x0
p1,1
p2,1
pp
p2
Figure 2: Illustration of how the zone path is cre-ated.
Clearly does not correspond to time in prac-tice but the idea of
deceleration and accelerationalong the path can be used as an
interpretation ofthe process of interpolation. The index and
theindex lzj of zone j are related through
= 1 +lzj
l(j+1)zj, (2)
which gives [0, 2]. For a given and a given
2
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zone j = i the normalized index for each seg-ment, li and li+1,
are computed through integra-tion of dlid and
dli+1d . Two different choices of
these derivatives will be presented below.
2.3 Examples of zone path interpolationA fundamental requirement
when constructingthe zone path is that the resulting path should
becontinuous with continuous derivative and prefer-ably a
continuous second derivative. These re-quirements can be
transformed into requirementson the second derivatives d2li
d2and d
2li+1d2
.
To get a continuous second derivative of thepath at the
beginning and the end of the zone,the second derivatives of l
should be close to zeronear these points. Also, to minimize the
acceler-ation of the robot end effector along the path, thesecond
derivative of l should be kept small withinthe zone.
A constant second derivative of index l, withrespect to , gives
the first derivatives of index l asthe solid lines shown in Figure
3. This method ofzone interpolation will be referred to as Method1.
The values of the normalized indices li and li+1are given by the
integrations
li = 1 2
dlidx
dx, (3)
li+1 =
0
dli+1dx
dx . (4)
This means that for zone i, li [lizi, 1] and li+1 [0, l(i+1)zi].
As a result of the nonzero value ofthe second derivative of the
index at = 0 and = 2, the second derivative of the path, d
2p(lc)dl2c
,
will not be continuous at the beginning and theend of the zone.
A smaller discontinuity can beachieved with the derivative of the
index as thedash-dotted lines shown in Figure 3 and this
in-terpolation method will be referred to as Method2. The indices
are here calculated according to (5)and (6).
li = 1 2
((1 lizi)(1 x2 ) + (1 lizi)
12(cos(
pix2 ) + 1) (1 x2 )()
)dx
(5)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3: Example of linear (solid) and non-linear(dashed)
choice of d2li
d2and d
2li+1d2
(described in(5), (6), and (7)).
li+1 =
0
(l(i+1)zix
2+ l(i+1)zi
12(cos(
pix2 + pi) + 1) x2()
)dx
(6)
Here () is constructed as two connected secondorder
polynomials.
() ={k2 + 1 when 1k( 2)2 + 1 when > 1 (7)
A small value of k in (7) gives a deeper zone pathwith smaller
discontinuities, but with a larger sec-ond derivative. By changing
the function (),different zone paths can be created.
3 The path in joint spaceThe next step is to transform the path
in Cartesianspace into a path in joint space. The path in
jointspace is here supposed to be represented using cu-bic
splines.
3.1 An algorithm for path generation injoint space
The general algorithm is given by Algorithm 1(from [7, 8]).
Algorithm 1 requires that index lc is used. Asdescribed in
Section 2.3, index lz , which is basedon the normalized index l [0,
1] is used in the
3
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Algorithm 1 Joint path interpolation.n = number of breakpoints
in each splinelc = 0hlc = distance between breakpointswhile not end
of section do
for i = 0 to n 1 doCompute pi for index lc + hlci and com-pute
the corresponding joint angle qi withinverse kinematics.
end forCalculate dq0dlc and
dq(n1)dlc
.
Compute the complete cubic spline q(lc).Adjust the spline
interpolation length basedupon the computed error P(lc)P(lc).
end while
interpolation of the zone paths. The relationshipbetween index
lc and lz can be approximated by
lc =lzlz
P(lz + lz)P(lz) . (8)
Since no inverse function of (8) can be found eas-ily, this
equation also, by iterations, gives the so-lution to an index lz
for a specific lc. In order forthe transformation to succeed, it is
important thatthe approximation error in (8) is kept small.
In Algorithm 1 the maximum deviation of theactual path from the
desired path is used to eval-uate each computed spline. This
deviation canbe approximated with the deviation in betweeneach two
adjacent breakpoints of the spline func-tion, i.e. points given by
index lc + hlc2 , . . . , lc +(2n 3)hlc2 . The joint angles
correspondingto these indices are transformed into Cartesianspace
using forward kinematics, and then com-pared to the desired
Cartesian path. To computethe splines in joint space a complete
spline isused, see e.g. [5].
3.2 Adjustment of the length of the splineIn order to keep a
certain level of path accuracy,the length of each spline must be
adjustable. Thisadjustment can be made in many ways. The com-plete
cubic spline used here, has a maximum error
given by
maxxixxi+1
|s(x) f(x)| < 5384
Mh4 , (9)
(see e.g. [5]) where the function f is assumed4 times
continuously differentiable on [xi, xi+1]and
|f (4)(x)| M , xi x xi+1 . (10)
h is the maximum length of interval [xi, xi+1].Using the upper
bound of the error in (9) algo-rithms, such as Algorithm 2, for
adjusting thelength of the spline can be constructed.
Large values of and in Algorithm 2 resultin a smaller value of
h, i.e. a shorter length be-tween the breakpoints, when the current
deviationis close to the specified maximum value, .
Algorithm 2 Adjust the spline length. = P(lc) P(lc) = maximum
value of x = /if x > 1 then
i = x1/4
if i > thenh = h i
elsekeep h
end ifelse
r = 1x1/4if r > thenh = h r
elseh = h
end ifend if
4 An example
The algorithms presented in this paper will nowbe applied to a
Cartesian line-circle path. A three-link robot with the Denavit
Hartenberg [10] pa-rameters according to Table 1 will be used.
Formore details on the robot and the kinematics see[8].
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Link/joint i ai i dii [rad] [m] [rad] [m]1 pi/2 0.15 0 0.4752 0
0.6 pi/2 03 pi/2 0.72 0 0
Table 1: Denavit Hartenberg parameters for athree-link
robot.
The line is defined by a start and end point, p1and p2, while
the circle is defined by a start point,p2, an intermediate point,
p3, and an end point,p4. The points have the following
coordinates:
p1 =(1 0.02 0.5
)p2 =
(0.7 0.2 0.5
)p3 =
(0.85 0.35 0.5
)p4 =
(0.74 0.49 0.5
) (11)The segments, together with the two differenttypes of zone
paths, can be seen in Figure 4.The dotted zone path is generated
according tointerpolation method 1 and the solid zone pathis
generated with method 2. The path is next
0.650.7
0.750.8
0.850.9
0.951
00.1
0.20.3
0.40.5
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
xy
z
Figure 4: The line-circle path with the two typesof zone
paths.
transformed into cubic spline functions in jointspace (Algorithm
1) using two breakpoints in ev-ery spline function [8]. The maximum
alloweddeviation error is set to 105 meters. The resultsof the
transformation can be seen in Figures 5to 7. In Figure 5 the knots
for the splines gen-erated by the algorithm in joint space is
shownwhen the zone is generated according to interpo-lation method
1, a similar result but with somemore knots is achieved for method
2. In Figure 6
the resulting d2q(lc)dl2c
using the two zone interpola-tion methods from Section 2.3 is
shown. Method1 has a lower peak value of the second derivativewith
respect to lc while method 2 gives a continu-ous second derivative.
In Figure 7 the path error isevaluated for the two different
interpolation meth-ods and it is shown that the error is kept
inside theupper bound, 105 meter, along the path in bothcases.
Notice that since the interpolated path isdifferent in the zone the
total length of the path isdifferent in the two cases.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.71.5
1
0.5
0
0.5
1
1.5
2
2.5
lcq(lc)
Figure 5: Location of breakpoints for the splinesin joint space
for the zone interpolation accordingto method 1.
5 ConclusionsThe idea of dividing the trajectory genera-tion
problem into two sub-problems, pathplanning/generation and dynamic
plan-ning/optimization, is presented. The firstproblem is solved
and is shown how it is possibleto create a general path using lines
and arcs. Thegeneration of a smooth path in Cartesian space
isdiscussed and two different approaches to zoneinterpolation are
presented. An algorithm forpath generation in joint-space is
formulated basedupon the idea of representing the joint path
withcubic splines. Finally an example shows how aCartesian path can
be generated and transformedinto a spline representation in
joint-space.
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(1985). Time-optimal control of robotic ma-nipulators along
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5
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7120
100
80
60
40
20
0
20
40
60
80
lc
(a) Method 1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7150
100
50
0
50
100
lc
(b) Method 2.
Figure 6: d2q(lc)dl2c
for the line-circle path using thetwo zone interpolation methods
from Section 2.3.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 105
lc
(a) Zone interpolation method 1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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lc
(b) Zone interpolation method 2.
Figure 7: Evaluation of path error, P(lc) P(lc), for line-circle
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