1. science
2. mathematics
3. algebra

Procedia Manufacturing
Volume XXX, 2015, Pages 1–10
43rd Proceedings of the North American Manufacturing Research
Institution of SME http://www.sme.org/namrc
Exponential and Sigmoid-Interpolated Machining
Trajectories
Christopher DiMarco1*, John C. Ziegert2†, and Christopher Vermillion3‡
1,2,3
The University of North Carolina at Charlotte, Charlotte, NC, USA.
cdimarc1@uncc.edu, jziegert@uncc.edu, cvermill@uncc.edu
Abstract
In single-point metal turning and boring processes, a chip nest can often be created that is a hazard
to part and operators alike. In order to mitigate this, a process called modulated tool path (MTP)
machining was developed that superimposes a sinusiodal motion tangent to the feed direction onto the
tool feed path to break chips. The sinusoidal motions are created under CNC control in the part program.
In the current implementation, the sinusoidal motion is approximated as a series of short linear moves.
Linear Interpolation is currently used to create position and velocity commands to the axis servomotors
at each control loop closure. Linear interpolation is a computationally heavy and dated method that is
not well tailored to a sinusoidal trajectory. In this paper a new method called the sigmoidal interpolator
is introduced that honors all physical constraints of a machining system while offering better tracking
performance and lower accelerations than the linear interpolator, all while reducing the number of
possible state transitions of the implemented software from approximately 17 to 4.
Keywords: Machining, Trajectory Planning, Tool Path Generation, Interpolation Techniques
1 Introduction
Metal turning and boring processes can often lead
to a 'chip nest,' or a long strand of metal that can wrap
around the tool tip or part. A chip nest can damage the
part finish and also presents a safety hazard, as machine
operators removing the nest can be injured by the razorsharp strand (see Figure 1). One solution to this issue
Figure 1: A chip nest forming on a machine
is called Modulated Tool Path (MTP) machining, a
process by which the tool tip is oscillated in the tool during a turning operation. After a few
moments of cutting, these razor sharp strands
become tangled with the tool and workpiece.
*
Graduate Research Student, Center for Precision Metrology
Instructor, Center for Precision Metrology
‡
Instructor, University of North Carolina at Charlotte
†
Selection and peer-review under responsibility of the Scientific Programme Committee of NAMRI/SME
c The Authors. Published by Elsevier B.V.
1
Exponential and Sigmoid-Interpolated Machining Strategies
Dimarco, Ziegert, and Vermillion
Position (units)
direction of the instantaneous tool feed motion. A new chip is formed each time the tool enters and exits
the cut (Woody, et al., 2012).
In a single-axis tool path where a normal cutting motion is a position versus time ramp, the oscillation
is added by superimposing a sinusoid on the ramp. The frequency and amplitude of the oscillation, as
well as the spindle speed, control the length of the chip that is removed and can also have an impact on
the surface finish of the part (Berglind & Ziegert, 2013).
The NC part program generally only provides the start and end coordinates of linear moves along
with the desired feed rate during that move. The servo-control system requires a position and velocity
to be specified at each closure of the control loop. These intermediate points and velocities are generated
by an interpolation algorithm built into the controller. There are many interpolator implementations.
When smooth and vibration-free motions are desired, it is common to use an algorithm that places limits
on allowable jerk, while simultaneously respecting the physical acceleration and velocity capabilities of
the machine. This algorithm is outlined by Altintas (Altintas, 2012). When it is desired to maximize
fidelity to the nominally commanded path, the jerk curve is rectangular with only three values for jerk,
j(t): ±jlim, or zero. This in turn leads to a trapezoidal-shaped acceleration profile with no discontinuities
that would require instantaneous changes in the force or torque. The velocity and position profiles take
on parabolic and cubic characteristics respectively. A maximum acceleration may also be instituted,
but often it is directly evaluated from the maximum allowable jerk (Altintas, 2012). A sample singleaxis trajectory is shown below in Figure 2.
Linearly Interpolated Trajectories
4.5
The linear interpolator is the subject of
a significant body of literature.
4
Researchers seeking to improve machine
3.5
positioning performance have investigated
advanced controller design (Koren & Lo,
3
1992), improving contour accuracy (Lee,
2.5
2005) (Ramesh, et al., 2005), input
2
shaping (Peleaz, et al., 2004), and high
performance (Kim & Chung, 2005). While
1.5
the linear interpolator is conceptually
1
simple, practical implementation relies on
Interpolated Trajectory
0.5
a large network of if/then statements to
Target Trajectory
determine at what times during each move
0
0
0.2
0.4
0.6
0.8
1
to switch from one jerk value to another in
Time (s)
order to assure the axis reaches the desired
Figure 2: A sample linear interpolated movement. The green
future positions and velocities without
overshoot. This requires approximately 17 line is the desired movement as provided by the NC program. The
different branching conditions for all blue curve is the jerk-limited linear interpolator output.
possible states of position, velocity, acceleration, and jerk. Since branch conditions are the most
expensive processes in terms of computing time, this invariably leads to large computation time costs.
Secondly, there is often an interpolator delay created by the acceleration and jerk limitations that is never
recouped. In Figure 2, this is readily seen by the growing time differential between the desired curve
(green) and the interpolated curve (blue). Typical linear interpolator implementations accept that the
idealized motions embodied in the part program are physically unrealizable because the instantaneous
velocity changes would require infinite acceleration, and make no attempt to bring the actual tool
trajectory back onto the idealized motion when away from the transients at the beginning and end of
each segment.
This paper reviews strategies for jerk and acceleration limited trajectory planning called the
exponential interpolator developed by Rymansaib (Rymansaib, et al., 2013), and proposes a sigmoidal
interpolator to replace the planning previously done by linear interpolators at transitions. This technique
2
Exponential and Sigmoid-Interpolated Machining Strategies
Dimarco, Ziegert, and Vermillion
will be shown to be more appropriate for periodic inputs, as well as being useful for standard machining
processes, and requiring fewer branching conditions.
2 Sigmoidal Interpolators
Conceptually, the proposed interpolator works by multiplying the nominal motions described in the
NC part program by a sigmoidal, or S-shaped, “blending” function that smoothly transitions from zero
to one, one to zero, or zero to one and back to zero. The result is a continuously differentiable curve
that obeys the initial conditions of the commanded motion and reaches the desired nominal trajectory
when the value of the function reaches one. Many sigmoid-type functions exist, including the logistic
function, the arctangent, the hyperbolic tangent, and the error function. Slightly different
implementations are used for the case of startup from rest, and transition from one commanded velocity
to another.
2.1 Startup
For Modulated Tool Path machining, each motion segment is composed of a linear component for
nominal feed and a superimposed sinusoidal component for chip breaking. Making use of superposition,
these two components can be evaluated separately and then summed to get position, velocity,
acceleration, and jerk values.
Linear Component. The linear component of the exponential interpolator is detailed here, as well as
by Rymansaib (Rymansaib, et al., 2013). The baseline path for the interpolator to match is given by the
linear equation in (1) below, where x is the resulting position and m is the linear slope:
x = mt + b
(1)
For initial startup, the equation for the sigmoid interpolator, SL is:
3
SL = 1 − e−αt
(2)
where α is a time constant, and t is the time. The order of the time variable t is chosen as 3 to maintain
a zero value at t=0 for acceleration, as will be shown below. This leads to a sigmoid interpolated position
value xL of:
3
xL = xSL = (mt + b)(1 − e−αt )
(3)
Note that at t = 0 and for sufficiently large values of t, x L= x. The velocity, ẋL, is:
3
𝑥̇ 𝐿 = m − e(−αt ) (m − 3αmt 3 − 3αbt 2 )
(4)
The velocity is equal to 0 at t=0, and asymptotically approaches the commanded feedrate, m. The value
of α determines how quickly this occurs. Differentiating a second time, the acceleration, ẍL, is equal to:
3
𝑥̈ 𝐿 = e−αt (−9α2 b𝑡 4 − 9α2 m𝑡 5 + 6αbt + 12αmt 2 )
(5)
Again, at t=0, the acceleration is equal to zero. With ẍL limited to a physically realizable maximum
acceleration and the values of t known for the movement, the maximum allowable value for α can be
determined using a numeric solver.
3
Exponential and Sigmoid-Interpolated Machining Strategies
The linear paths generated by the
interpolator are shown below in Figure 3.
Note that the sigmoidal interpolator causes
the feedrate of the tool to exceed the
desired value for some period of time and
it tries to “catch up” to the commanded
trajectory. As α increases, the planned
trajectory more quickly reaches the
desired path.
Dimarco, Ziegert, and Vermillion
Sigmoidal Interpolator - Linear Component
10
9
8
Position (units)
7
6
5
4
3
Sinusoidal Component. The desired
sinusoidal motion is given by:
Target
 =1
 =3
 =5
2
1
x = A sin(ωt)
(6)
0
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
where A is the sine wave amplitude and ω
Figure 3: The linear exponential component of the
is the angular frequency. As with the linear
interpolator.
As α increases, the trajectory more rapidly
component, the sinusoidal component is
approaches the desired path (the black dashed line).
multiplied by a sigmoidal function to
control velocity and acceleration. The form is similar:
2
SS = 1 − e−αt
(7)
However, the order of t is decreased to 2 as it is not required to be increased to maintain zero acceleration
at time step zero once coupled with the sinusoid. Multiplying these together yields the position equation
for the sinusoidal component:
2
xS = xSS = (A sin(ωt))(1 − e−αt )
(8)
Differentiating twice yields the velocity and acceleration equations, respectively:
2
2
𝑥̇𝑆 = 2Aαte−αt sin(ωt) − Aωe−αt cos(ωt) + Aω cos(ωt)
2
2
𝑥̈𝑆 = −𝐴𝑒 −𝛼𝑡 [4𝛼 2 𝑡 2 sin(𝜔𝑡) + 𝜔2 𝑒 𝛼𝑡 sin(𝜔𝑡) − 2𝛼 sin(𝜔𝑡) − 4𝛼𝜔𝑡𝑐𝑜𝑠(𝜔𝑡) − 𝜔2 sin(𝜔𝑡)]
(10)
At t=0, both velocity and acceleration are
zero. As with the linear component, since all
values other than α are known, a maximum
feasible can be computed numerically ensuring
that the resulting trajectory does not exceed
acceleration or velocity limits. An example of
the interpolated trajectory for α values from 1
to 5 is shown in Figure 4.
Sigmoidal Interpolator - Sinusoid Component
0.2
0.15
0.1
Position (units)
(9)
0.05
0
-0.05
Target
 =1
 =3
 =5
-0.1
-0.15
-0.2
0
0.5
1
1.5
2
Time (s)
2.5
3
The linear component and sinusoidal
components are combined to create an MTP
path, as shown below in Figure 5.
3.5
Figure 4: The sinusoidal exponential component of the
interpolator, shown with three α values.
4
Exponential and Sigmoid-Interpolated Machining Strategies
Dimarco, Ziegert, and Vermillion
2.2 Sigmoid Interpolator at Transitions
At a transition point where a change in
the feed velocity or direction is desired, a
different approach can be used since the
desired trajectory is a continuous curve.
The interpolator utilizes two sigmoids to
create a composite function that sweeps
smoothly and asymptotically from 0 to 1
and back again to zero. The sigmoid
equation is:
Sigmoidal Interpolator - Combined
10
9
8
Position (units)
7
6
5
4
3
Target
 =1
 =3
 =5
2
1
S=
1
1
(11)
+ e−α(t−c)
where c is the transition point (in time)
when the function equals one, and α is
again the time constant that controls how
Figure 5: Combined linear/sinusoidal exponential fast the transitions occur. A positive α is
components. Higher α more rapidly approaches the desired path used for the increasing portion of the curve
(black dashed line).
and a negative one for the outgoing. At
t=c, both curves contribute exactly half of their total value to the position at c, guaranteeing the correct
transition point is reached. Unfortunately, this does not ensure a zero velocity at the transition point,
which is imperative for direction switching. To counteract this a correction term must be solved for and
added to the sigmoid:
0
0
0.5
1
1.5
2
Time (s)
2.5
3
Sin =
Sout =
3.5
𝑎
(12)
a+e−α(t−c)
𝑏
𝑏 + e−α(t−c)
(13)
where a and b are the incoming and outgoing correction coefficients, respectively. The input and output
trajectories each have a separate correction term that is derived from
twoCrossover
constraints
Sigmoid
@ c = to
5 be enforced on
1
the transition. Firstly:
0.83
0.9
a-Sigmoid
(14)
S𝑖𝑛 (c) + S𝑜𝑢𝑡 (c) = 1
3
0.8
b-Sigmoid
Transition Time
0.7
Sigmoid Value
Values of a and b can be chosen to shape
the blending of the incoming and outgoing
curves, where a = b = 1 will reflect an even
blending throughout the transition, a > b will
reflect a late transition, and b > a will reflect
an early transition. In particular, values of a
and b can be chosen by first selecting the
desired blending ratio at the transition
midpoint c as R = Sin^des(c) / Sout^des(c).
Since Sin (c) = a/(1+a) and Sout (c) =
b(1+b), we can achieve Sin(c) / Sout(c) = R
by selecting a and b as shown below:
0.6
0.5
0.4
0.16
7
0.3
0.2
0.1
0
0
2
4
6
8
10
Time (s)
Figure 6: Corrected Sigmoid Values at Transition
– The blue curve is the value of the incoming segment
and the green curve is the outgoing segment.
5
Exponential and Sigmoid-Interpolated Machining Strategies
Dimarco, Ziegert, and Vermillion
𝑎=𝑅
𝑏=
(15)
1
(16)
𝑅
Note that these values are unaffected
by
α, and that velocity and acceleration
Incoming Trajectory
5
may increase even if the outgoing
Outgoing Trajectory
Sigmoid Curve
trajectory is decreasing. Also, it is
4
unnecessary to perform this correction on
3
trajectories continuing in the same
2
direction. An example of a sigmoid
1
calculated in this manner is shown in
Figure 6. Using these Sin and Sout values, an
0
acceleration and jerk-limited curve passes
-1
smoothly through the transition point and
-2
does not overshoot or gouge the part in any
-3
portion of the transition, as shown below
in Figure 7.
-4
0
2
4
6
8
10
As for the sinusoidal component of the
Time (s)
path, care must be taken so that the
Figure 7: Sigmoid transition from 1:5 slope to -1:1 slope. At
time t=5, the transition has achieved its target point and does not oscillation does not gouge the part when
nearing transition points. To prevent this,
exceed it.
using (12) and (13) with a c value away
from the transition point coupled with an a or b value that is very small (but not zero) will remove the
sine component from the transition trajectory. This method can also be used to end a move.
Sigmoid Transition
X Position (units)
6
3 Results
3.1 Exponential Interpolation of MTP Paths
As shown in the figures below, the two methods of trajectory interpolation produce vastly different
results, especially at low acceleration limits. Figures 8-11 show that tracking is similar between the two
at high acceleration limits, but a time delay still exists from the intended path.
Linear Interpolator - High Acceleration Limit
Linear Interpolator - Low Acceleration Limit
3.5
4
Linear Interpolator
Target Trajectory
3.5
3
2.5
Position (units)
Position (units)
3
Linear Interpolator
Target Trajectory
2.5
2
1.5
2
1.5
1
1
0.5
0.5
0
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
Figure 9: Linear Interpolator Trajectory with a low
acceleration limit.
6
0
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
Figure 8: Linear Interpolator Trajectory with a
high acceleration limit.
Exponential and Sigmoid-Interpolated Machining Strategies
Sigmoid Interpolator - High Acceleration Limit
Sigmoid Interpolator - Low Acceleration Limit
3.5
3.5
Target Trajectory
Sigmoid Interpolator
3
3
Target Trajectory
Sigmoid Interpolator
2.5
Position (units)
2.5
Position (units)
Dimarco, Ziegert, and Vermillion
2
1.5
2
1.5
1
1
0.5
0.5
0
0
0.5
1
1.5
2
Time (s)
2.5
3
0
0
3.5
Figure 10: Sigmoid Interpolator with a low
acceleration limit.
0.5
1
1.5
2
Time (s)
2.5
3
3.5
Figure 11: Sigmoid Interpolator with a high
acceleration limit.
3.2 Quantitative Performance Analysis
In order to provide a fair, quantitative performance comparison of the baseline linear interpolator
and the new sigmoidal interpolator, we defined a performance index, J, comprised of two terms, namely
the tracking error and the tool acceleration. With the main concern being tracking error, a high value
was placed on properly following the desired trajectory. A secondary concern was the acceleration of
the system. This lead to a total performance index of the form:
𝑇
𝐽 = ∫0 [𝑘𝑒 (𝑦𝑑𝑒𝑠 − 𝑦𝑎𝑐𝑡 )2 + 𝑘𝑎 𝑎𝑐𝑐𝑒𝑙 2 ] 𝑑𝑡
(17)
Cost Function Value (Tracking Error2)
where T is the run time, ydes is the position to achieve, yact is the interpolated value passed to the controller
at each loop closure, and accel is the instantaneous acceleration value. ke and ka are the weighting
Linear Interpolator vs. Sigmoid Interpolator - Tracking/Acceleration
coefficients for the error tracking and acceleration
3000
costs, respectively. A value of 1.7x107 was
Linear Interpolator
Sigmoid Interpolator
chosen for ke, and ka was set to 1. The data below
2500
was generated for a ramped sinousoid of slope 3,
amplitude 0.15, and frequency of 2Hz.
2000
1500
1000
500
0
0
50
100
150
200
250
300
Acceleration Limit (units/s2)
Figure 12: Tracking Error – Linear vs. Sigmoid
Interpolator. The sigmoid interpolator is the lower green
curve, the linear interpolator is the upper blue curve.
Error Tracking. Taken separately, the error
tracking showed an improvement when compared
to the linear interpolator over a range of
acceleration limits. The exponential/sigmoidal
interpolator has a significant advantage in error
tracking at low acceleration limits; but as that
value is increased, both interpolators tend toward
similar cost values. The results are shown in
Figure 12.
7
Exponential and Sigmoid-Interpolated Machining Strategies
Dimarco, Ziegert, and Vermillion
Achieved Acceleration ((max-achieved)2)
6
Acceleration Costs. As expected, both
Linear vs. Sigmoid - Acceleration
x 10
4
interpolators performed similarly with respect
Linear Interpolator
to acceleration costs, as they are both tracking
3.5
Sigmoid Interpolator
similar curves. The sigmoid interpolator does
3
function slightly better as limits are increased,
however. This can be attributed to the fact that
2.5
the linear interpolator is always chasing a
2
maximum value of acceleration and jerk,
while the sigmoid is merely tracking what the
1.5
curve needs to be achieved. This leads to
1
continuous jerk profiles as opposed to the
positive/negative/zero profile used by the
0.5
linear interpolator. The result is shown below
0
in Figure 13.
0
200
400
600
800
1000
When these two indices are combined and
Acceleration Limit (units/s2)
corrected with the ke and ka values, the result
Figure 13: Achieved acceleration. The sigmoid curve is
is shown in Figure 14 and paints a clear picture
the lower, more efficient green curve.
of the improved performance of the sigmoid
interpolator.
6
Linear vs. Sigmoid - Combined Performance
Combined Performance - (error2+accel2)
4
x 10
Alpha Value. The α value for the sigmoidal
interpolator determines how quickly the planned
trajectory approaches the desired path. For any
particular path, an alpha value can be chosen
that minimizes the cost of the movement.
Plotting total cost vs. α value gives the curve
shown below in Figure 15. The green line shows
the cost associated with the linear interpolator
for the same movement. As shown here, the
minimized cost is less than a third of that of the
linear interpolator.
Linear Interpolator
Sigmoid Interpolator
3.5
3
2.5
2
1.5
1
0.5
0
0
200
400
600
800
1000
Sigmoid Transition. At transition points, the
sigmoid blend also performed well. For a back
Figure 14: Combined tracking error and
and forth movement of equal distances and
acceleration achieved values for both interpolators.
speeds (10 inches and 10 in/s, respectively) the
linear interpolator displayed its two main
problems: tracking error, and not reaching the intended position. With an acceleration max of 500 in/s,
the sigmoid’s maximum allowable α was 50. Using similar values of ke, and ka, the performance
difference was drastic. By this performance index, the sigmoidal transition was 30 times more efficient
at tracking the target trajectory and lowering overall acceleration. See Figures 16 and 17 for detail.
Acceleration Limit (units/s2)
4
Conclusion
In this paper, an alternative to the linearly interpolated trajectory planner was proposed. The
combined exponential and sigmoidal interpolator derived here has the capability to follow a prescribed
path, honor physical (position, velocity, acceleration, jerk) constraints, and handle transitions without
overshoot. It was compared against the linear interpolator in a single axis for a span of acceleration
limits. In terms of tracking error, the sigmoid interpolator showed a 60% reduction in error for low
8
Exponential and Sigmoid-Interpolated Machining Strategies
Sigmoid Performance vs. -Value
Dimarco, Ziegert, and Vermillion
Transitions - Sigmoid vs. Linear
785.14
10
9
785.12
8
785.1
7
785.08
Position (in)
Performance Value (error2+accel2)
Sigmoid Interpolator
785.06
785.04
6
5
4
3
785.02
2
785
784.98
Target
Linear Interpolator
Sigmoid Interpolator
1
0
X: 10
Y: 785
100
200
-Value
300
400
Figure 15: Sigmoid performance by alpha values
0
0
10
Position (in)
9.8
9.6
9.4
9.2
Target
Linear Interpolator
Sigmoid Interpolator
0.8
0.9
1
Time (s)
1.1
1.2
1
Time (s)
1.5
2
Figure 16: Sigmoid Transition vs. Linear Interpolator
Transitions
Transitions - Sigmoid vs. Linear
9
0.5
1.3
Figure 17: Sigmoid Transition vs. Linear Interpolator close-up
acceleration maximums. As the limit of
acceleration was increased, however,
sigmoid interpolator error trended towards
a similar value as the linear interpolator.
Achieved acceleration was similar for both
interpolators, at low acceleration limits,
but as acceleration limits increase the
sigmoid interpolator becomes increasingly
more efficient, showing a 12%
improvement in performance at 1000
units/s2 acceleration. When these metrics
are combined it can be shown that the
sigmoid interpolator has a better
performance index across the tested range
of acceleration limits. Using the sigmoid
interpolator at a transition point with the
same maximum acceleration limit (and
calculating an optimal alpha value) as the
linear interpolator showed a 30-fold
improvement in performance.
4.1 Continued Work
Future work in the exponential/sigmoidal interpolator can be done to improve transitions, to
coordinate multi-axis moves, and validate this model on a live system. The proposed interpolator scheme
has the potential to improve contouring accuracy and reduce cycle times.
9
Exponential and Sigmoid-Interpolated Machining Strategies
Dimarco, Ziegert, and Vermillion
References
Altintas, Y. (2012). Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations,
and CNC Design, 1st Ed.. In: UK: Cambridge University Press, pp. 105-124.
Berglind, L. & Ziegert, J. (2013). Chip Breaking Parameter Selection for Constant Surface Speed
Machining. ASME 2013 International Mechanical Engineering Congress and Exposition Vol. 2B,
p. V02BTO2A039.
Kim, M. & Chung, S. (2005). A Systematic Approach to Design High Performance Feed Drive Systems.
International Journal of Machine Tools and Manufacture, V45, pp. 1421-1435.
Koren, Y. & Lo, C. (1992). Advanced Controllers for Feed Drives. CIRP Annals, Manufacturing
Technology V41 N2, pp. 689-698.
Lee, J. (2005). Design of Controllers for Improving Contour Accuracy in a High-Speed Milling
Machine. PhD. Dissertation, Dept. Mech. Eng., University of Florida.
Peleaz, G. et al. (2004). Input Shaping Reference Commands for Cartesion Machines. Control
Engineering Practice, V13, pp. 941-958.
Ramesh, R., Mannan, M. & Poo, A. (2005). Tracking and Contour Error Control in CNC Servo Systems.
International Journal of Machine Tools and Manufacture, V45, pp. 301-326.
Rymansaib, Z., Iravani, P. & Sahinkaya, M. (2013). Exponential Trajectory Generation for Point to
Point Motions. Wollongong, Austrailia, s.n., pp. 906-911.
Woody, B. et al. (2012). Methods and systems for chip breaking in turning applications using CNC
toolpaths. USA, Patent No. 8,240,434.
10