Feedrate Planning for a Delta Parallel Kinematics Numerically

Controlled Machine using NURBS Toolpaths

Gabriel Karasek

a

and Krystian Erwinski

b

Institute of Engineering and Technology, Faculty of Physics Astronomy and Informatics,

Nicolaus Copernicus University, Torun, Poland

Keywords:

Numerical Control, Delta Parallel Kinematics, NURBS, Feedrate Planning.

Abstract:

This paper presents a concept of a computationally efﬁcient feedrate planning algorithm for parallel kinematics

machine in a linear delta conﬁguration. Non-Uniform Rational B-Spline (NURBS) polynomial curve is used

for toolpath deﬁnition which provides a smooth trajectory. The feedrate proﬁle is deﬁned as a jerk limited S-

Curve which takes into account limitations stemming from the toolpath curvature and the machine kinematics.

The article presents the outline of the method, structure of the experimental station including the real-time

control system and preliminary experimental results. Further direction of the research is also described. The

proposed method can provide a smooth motion trajectory obtained in real-time with small computational

requirements.

1 INTRODUCTION

In numerical control feedrate is the end effector veloc-

ity tangent to the toolpath. It is the result of combin-

ing the velocities of all of the machine’s axes. Fee-

drate planning is the process of determining a fee-

drate proﬁle that provides a proper value of end ef-

fector feedrate while respecting the machine’s limita-

tions such as maximum velocity, acceleration and jerk

of its axes. These limitations usually stem from lim-

iations of the drive power, drivetrain capabilities or

contouring error.

The feedrate planning problem has been studied

by many researchers for cartesian machines utlizing

optimization algorithms with explicit inclusion of ax-

ial constraints (Mercy et al., 2018; Lu et al., 2018),

tracking error constraints (Zhang et al., 2019) or con-

tour error constraints (Chen and Sun, 2019). Alterna-

tively simpliﬁed approaches were proposed using the

feedrate limit function which combines axial limita-

tions of velocity, acceleration and jerk into a single

feedrate limit (Ni et al., 2018b; Ni et al., 2018a; Er-

winski et al., 2022). In the second case the feedrate

proﬁle is usually the widely used jerk-limited s-curve

proﬁle. These feedrate planning methods usually con-

sider cartesian machines sometimes with additional

a

https://orcid.org/0000-0001-8031-8323

b

https://orcid.org/0000-0001-6899-1785

rotary axes in a 5-axis conﬁguration (Li et al., 2021).

Furthermore many of the aforementioned methods

utilize the Non-Uniform B-Spline polynomial curve

as a state-of-the-art way do deﬁne the toolpath which

can provide a smooth trajectory compared to typical

G-Code.

The feedrate planning problem for NURBS tool-

paths is more complex for machines with non-

cartesian kinematics which include parallel kinemat-

ics machines. This problem was not investigated pre-

viously as much for these types of machines but is

currently an emerging research topic (Su et al., 2018;

Li et al., 2021). Investigation of feedrate planning

with inclusion of joint/drive limitations is the main

focus of the authors’ current research. The delta par-

allel conﬁguration with linear actuators is the authors’

main focus as such machines can achieve large fee-

drates and therefore high productivity with more stiff-

ness than delta robots with rotary joints. Due to highly

non-linear dependency between joint and end effec-

tor velocities, accelerations and jerks the problem is

much more complex than for cartesian machines. The

inverse kinematics has to be taken into account if ax-

ial constraints are to be considered. Additionally the

authors’ objective is to consider the computational ef-

ﬁciency and implementation aspects which are impor-

tant if the developed feedrate planning methods are to

be used in practice.

In this paper a computationally efﬁcient method

Karasek, G. and Erwinski, K.

Feedrate Planning for a Delta Parallel Kinematics Numerically Controlled Machine using NURBS Toolpaths.

DOI: 10.5220/0011352900003271

In Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2022), pages 743-749

ISBN: 978-989-758-585-2; ISSN: 2184-2809

Copyright

c

2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

743

for feedrate planning is presented for a delta paral-

lel kinematics numerically controlled machine. The

jerk limited S-Curve proﬁle is used for feedrate plan-

ning and NURBS curves are used as toolpath deﬁ-

nition. The feedrate planning and NURBS curve in-

terpolation algorithms are described. The experimen-

tal station is described including a description of the

algorithms’ implementation in a real-time Linux op-

erating system. Preliminary experimental results of

the feedrate proﬁle and individual axial velocity, ac-

celeration and jerk proﬁles and following errors are

presented.

2 NURBS CURVE TOOLPATH

DEFINITION AND

INTERPOLATION

Non-Uniform Rational B-Splines (NURBS) are poly-

nomial spline curves commonly used in computer

graphics but also as deﬁnitions of complex shapes

for Computerized Numerically Controlled (CNC) ma-

chines. They are deﬁned in cartesian space using con-

trol points forming a control polygon which roughly

deﬁnes the curve’s shape. The shape can be further

inﬂuenced by control point weights which pull the

the curve closer to the point if the weight is above

1. NURBS are rational curves which means they can

be used to deﬁne most shapes. An additional advan-

tage is the ability to locally inﬂuence the shape of the

curve by moving each control point. Furthermore a

curve of a deﬁned degree exhibits continuity of it’s

derivatives if interior knots are not repeated. A cubic

curve has G2 geometric continuity which means that

it’s second derivatives are continuous. This ensures

smoothness of the curve as well as the motion trajec-

tory of the end effector moving along this curve. An

example of a NURBS curve is shown of Fig.1.

Mathematically NURBS curves are deﬁned as

(Piegl and Tiller, 1996):

C(u) =

n

∑

i=0

N

i,p

(u)w

i

·P

i

n

∑

i=0

N

i,p

(u)w

i

(1)

where: u - curve parameter value, N

i,p

(u) - value of

an elementary polynomial basis function, P

i

- con-

trol point coordinates, w

i

- control point weight. Each

point on the curve is identiﬁed by a value of the unit-

less parameter with a value between 0 and 1 which in-

dicate the beginning and end of the curve respectively.

In practice evaluation of the curve is performed us-

ing the numerically stable DeBoor’s algorithm. This

20

100

40

60

50

100

80

Z [mm]

100

50

Y [mm]

0

120

X [mm]

0

140

-50

-50

-100

-100

Figure 1: ”Bird” NURBS toolpath (blue) with marked con-

trol points and polygon (orange).

algorithm is also used to compute curve derivatives

with respect to the curve parameters.

Interpolation is the process of evaluating the tool-

path in constant time intervals to generate points on

the curve. In case of cartesian machines coordinates

of the generated points are the position setpoints of

each machine axis. In case of parallel kinematics ma-

chines the generated coordinates are transformed us-

ing inverse kinematics to generate setpoints for each

joint. In each interpolation step a new value of the

curve parameter u

i+1

is produced corresponding to

the desired arc-length increment and a point on the

NURBS curve C(u

i+1

) is produced using DeBoor’s

algorithm. Arc-length increments ∆s between consec-

utive interpolation points (u

i

,u

i+1

) should be propor-

tional to the desired feedrate F

i

in the current interpo-

lation step. This can be expressed as:

∆s =

Z

u

i+1

u

i

||C

0

(u)||du = V (t

i

) ·∆t (2)

where: C

0

(u) - ﬁrst derivative of the curve with re-

spect to the curve paramter, V (t

i

) - desired feedrate at

current interpolation time step, ∆t - interpolation time

step. This process is illustrated in Fig.2.

C(u

i

)

Δs = V(

i

) t

t

V

t

i

V(

i

)

FEEDRATE PROFILE

NURBS TOOLPATH

C(u

i+1

)

0

t

end

t

Δ

t

u=0

u=1

Figure 2: Illustration of NURBS interpolation process

based on a polynomial feedrate proﬁle.

The relationship between the curve parameter in-

crement ∆u = u

i+1

−u

i

and arc-length increment ∆S is

non-linear and unique for each curve. This means that

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

744

in each interpolation step this relationship has to be

approximated to satisfy the condition stated in eq.2.

There are several methods of NURBS interpola-

tion such as second and third order Taylor series ap-

proximation, Predictor-Corrector or Feed Correction

Polynomial (Erwinski et al., 2021). In this paper the

second order Runge-Kutta Predictor-Corrector (RK2-

PC) two-step method proposed in (Jia et al., 2017) is

used. In the ﬁrst step (predictor) a coarse approxima-

tion of the desired curve parameter value is produced:

˜u

i+1

= u

i

+

1

2

(F

1

+ F

2

)∆t (3)

F

1

=

du

dt

=

V (t

i

)

||C

0

(u

i

)||

F

2

=

V (t

i

)

||C

0

(u

i

+ F

1

∆t)||

(4)

In the second step (corrector) a correction increment

∆u

i+1

is determined which modiﬁes the coarse result

˜u

i+1

of the ﬁrst step. Ideally this should eliminate

the arc-length error so that the following equation is

satisﬁed:

||C( ˜u

i+1

+ ∆u

i+1

) −C(u

i

)|| = V (t

i

) ·∆t (5)

The end position is rewritten using ﬁrst order Taylor

expansion and substituted into equation 5. Then both

sides of the obtained formula are squared and after

some manipulations a quadratic equation is obtained:

a ·(∆u

i+1

)

2

+ b ·∆u

i+1

+ c = 0 (6)

a = ||C

0

( ˜u

i+1

)||

2

(7)

b = 2 ·C

0

( ˜u

i+1

) ·(C( ˜u

i+1

) −C(u)) (8)

c = ||C( ˜u

i+1

) −C(u)||

2

−V

2

(t

i

)∆t

2

(9)

The solution to this equation minimizes the interpo-

lation error. The ﬁnal corrected NURBS parameter

value is thus given by:

u

i+1

= ˜u

i+1

+

−b +

√

a

2

−4a ·b

2a

(10)

It can be seen that in the corrector step the RK2-PC

method uses feedback from a trial interpolation to de-

termine the actual interpolation error. The most com-

monly used 2nd and 3rd order Taylor series approxi-

mations are open-loop methods. They are susceptible

to interpolation error accumulation when large fee-

drates and toolpath curvatures are involved. They also

require computation of higher order NURBS deriva-

tives(Erwinski et al., 2021).

3 DELTA PARALLEL

KINEMATICS

Parallel kinematics is an alternative kinematics con-

ﬁguration, where the end effector is connected with

the drives through two or more independent joints.

Faster feedrate and lower weight of moving system

are the main advantages compared to traditional carte-

sian machines with prismatic joints connected in se-

ries. Also, unlike cartesian machines, there is no ac-

cumulation of positioning error from each axis. All

of axes are moving independently, which means that

the maximum position error is equal to the maximum

error in each axis. Lower weight of individual axes

allows to achieve a lot higher accelerations - up to

300G. (Bouri and Clavel, 2010)

Most popular types of parallel kinematics are ro-

tary delta, mostly used in pick-and-place machines,

and linear delta, which most common applications are

3d printers and robots. This paper will focus on linear

delta machine kinematics. In this type of machines,

linear movements of parallel axes are combined to al-

low movement of end effector in 3 dimensional space.

Main disadvantage of this kinematic conﬁguration is

greater number of mathematical equations, that need

to be solved in order to determine the required joint

setpoints for a desired cartesian position of the end

effector (inverse kinematics).

Linear joint position (J

i

)

Figure 3: Effector arm of the delta kinematics machine.

Each of the three vertical linear joints is spread at

an angle of 120 degrees forming an equilateral trian-

gle with the origin in the center equidistant to each of

the vertical joints. As shown in Fig.3, position in X

and Y axes is determined by the size of the effector

plate and arm angle. Z position is dependant on the

height of effector plate above the zero-plane. Usually

it is also required to include end effector or tool off-

set in the Z axis. The inverse kinematics equation for

Feedrate Planning for a Delta Parallel Kinematics Numerically Controlled Machine using NURBS Toolpaths

745

each linear joint is deﬁned by the following formulas:

J

i

= Z +

q

L

2

−(X −r ∗C

i

)

2

−(Y −r ∗S

i

)

2

+ O

(11)

C

i

= cos(α

i

), S

i

= sin(α

i

) (12)

where: J

i

- linear joint positions, X,Y,Z - cartesian

coordinates of end effector position, r - arm radius

(distance between end effector plate and linear joint

carriage in the x,y plane), L - arm length, O - vertical

tool offset, i = A,B,C - linear joint index, α

i

- linear

joint angle (α

A

= 0

◦

,α

B

= 120

◦

,α

C

= 240

◦

)

4 S-CURVE FEEDRATE

PLANNING

The jerk limited feedrate proﬁle is one of most com-

monly used types of proﬁling in cnc control systems.

A full proﬁle consist of seven phases which are shown

on ﬁg.4. Depending on the distance between start and

F

start

F

max

F

end

A

max

-A

max

J

max

-J

max

0

0

Feedrate

Acceleration

Jerk

T

1

T

2

T

3

T

6

T

5

T

4

T

7

Figure 4: Example feedrate, acceleration and jerk proﬁles.

T

1

−T

7

are the durations of each phase of the proﬁle.

end point, start and end velocities (V

start

,V

end

), maxi-

mum feedrate, acceleration and jerk (V

max

,A

max

,J

max

)

the number of phases may be reduced. The algorithm

to determine the duration of each phase is given be-

low.

The duration of each phase of the S-curve proﬁle

is characterized by:

T

1

= T

3

= T

5

= T

7

=

A

max

J

max

(13)

T

2

=

V

max

−V

start

A

max

−T

1

(14)

T

6

=

V

5

−V

6

A

max

(15)

If A

2

max

/J

max

≥V

max

−V

start

the total increase in fee-

drate in phases 1 and 3 is equal or greater than the dif-

ference between start and maximum feedrate. Phase

2 is omitted and acceleration is triangular instead of

trapezoid. Analogously if A

2

max

/J

max

≥ V

max

−V

end

phase 6 is omitted. If phases 2 or 6 are omitted their

duration is 0 and:

T

1

= T

3

=

r

V

max

−V

start

J

max

(16)

T

5

= T

7

=

r

V

max

−V

end

J

max

(17)

After determining the duration of phases 1,2,3,5,6,7

using the formulas given above the distance covered

with the current proﬁle S

tot

is computed. If the dis-

tance is lower than the curve segment arc-length S

seg

for which the proﬁle is generated, phase 4 is added.

The duration of phase 4 with constant maximum fee-

drate of V

max

is T

4

= V

max

/(S

seg

−S

tot

). Otherwise if

the distance S

seg

−S

tot

is negative the maximum fee-

drate has to be lowered and becomes the peak fee-

drate V

peak

. The value of V

peak

has to be determined

numerically. Bisection search is performed between

V

a

= V

max

and V

b

= max(V

start

,V

end

) to ﬁnd the value

of V

peak

for which the total distance is just below the

segment distance. In each iteration the feedrate pro-

ﬁle duration and distance are recomputed with V

peak

instead of V

max

until S

seg

−S

tot

< V

peak

·∆t.

When the NURBS toolpath exhibits areas of sig-

niﬁcant curvature the axial velocities, accelerations

and jerks may vary signiﬁcantly even if a constant

feedrate is used. This is especially true in case of par-

allel kinematics machines. In order to decrease this

effect a feedrate limit function can be utilized. Such

a function combines axial velocity, acceleration and

jerk limits into a single feedrate limit. Furthermore

chord error limits can also be included. The chord

error is the distance between an arc between two con-

secutive interpolation points and a line between those

points. This approximates the interpolation error due

to discrete interpolation of the curve. The feedrate

limit function (FLF) can be expressed as (Sun et al.,

2019; Zhao et al., 2013):

V

lim

chr

=

2

∆t

q

ρ

2

(u) −(ρ(u) −ε

max

)

2

(18)

V

lim

acc

=

s

A

max

κ(u)

(19)

V

lim

acc

=

3

s

J

max

κ

2

(u)

(20)

κ(u) =

1

ρ(u)

=

||C

0

(u) ×C

00

(u)||

||C

0

(u)||

3

(21)

FLF = min

V

max

,V

lim

chr

,V

lim

acc

,V

lim

acc

(22)

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

746

where: κ - curvature, ρ - curvature radius, ε

max

- max-

imum allowable chord error.

The FLF is used to determine critical points of the

NURBS curve toolpath for which the feedrate has to

be reduced taking into account the maximum allow-

able chord error, feedrate, centripetal acceleration and

jerk. The NURBS curve is evaluated at constant in-

tervals of the u parameter and the value of the FLF is

computed at these points. The array of FLF value is

scanned do determine coarse local minima of the FLF.

A local minimum is a point for which its two neigh-

bouring points have a higher value of the FLF. These

coarse points are then reﬁned using Brent’s method

to ﬁnd a precise value of the FLF minimum and its

corresponding value of parameter u. The process is

performed for the whole curve and an array of curva-

ture critical points is generated. The S-Curve feedrate

proﬁle is then planned between these critical points

with the FLF values serving as initial and ﬁnal fee-

drates. Cartesian axis limits are also applied by it-

eratively repeating the interpolation process with the

s-curve segment and verifying if axial constraints are

violated. If that is the case maximum acceleration

and/or jerk maximum values are decreased and the

proﬁle is generated again. This is repeated until all

axial constraints are satisﬁed. A detailed description

of the algorithm used for a cartesian machine is given

in (Erwinski et al., 2022). The algorithm will be fur-

ther improved to include additional constraints, espe-

cially linear joint velocity, acceleration, constraints.

This will require inclusion of the inverse kinematics

in the feedrate planning process.

5 EXPERIMENTAL STATION

In order to perform real-time control and gather ex-

perimental data a PC-based control system for linear

delta 3D printer was developed. A desktop PC based

controller was used (Intel i3-6100T@3.2GHz, 8GB

RAM) with Linux operating system (DEBIAN 11)

with the kernel patched using the RT-Preempt real-

time patch. This patch enables full kernel preemp-

tion which enables the real-time control processes to

run at the highest priority. LinuxCNC open-source

machine controller software, was installed to utilize

its real-time APIs (HAL and RTAPI). This allowed

to include custom feedrate planning and NURBS in-

terpolation algorithms and run them in the real-time

system. In order to interface the drives EtherCAT

real-time communication bus stack was used based on

the IGH EtherCAT master stack. Nanotec C5E servo-

stepper drives were used with Nema17 stepper motors

equipped with incremental encoders (4000 pulse/rev).

The drives and motors worked in closed loop vec-

tor control mode. All devices were connected via

Ethercat bus. CiA 402 state machine was used. Ad-

equate software driver, that allowed for a cycle time

of 250μs, was developed. Aforementioned driver also

allowed to use HAL API with Nanotec drives. The

block schematic of the experimental station is shown

in Fig.5.

NURBS

TOOLPATH

FEEDRATE

PROFILE

GENERATION

ALGORITHM

PARAMETERS

CONSTRAINTS

MEASUREMENT

DATA AQUISITION,

FILE STORAGE AND

VISUALIZATION

NURBS

TOOLPATH

INTERPOLATION

MODULE

SERVODRIVE

CONTROL

MODULE

CIA402

DEVICE

PROFILE

ETHERCAT

MASTER

STACK

REAL TIME

ETHERNET

DRIVER

REAL-TIME LINUX ENVIRONMENT

SHARED MEMORY

BUFFERS

NON-REAL-TIME PROGRAMS

REAL-TIME

THREAD

(250µs)

CONTROL WORD

POSITION COMMAND

DIGITAL OUTPUTS

STATUS WORD

ACTUAL POSITION

DIGITAL INPUTS

POSITION ERROR

ACTUAL VELOCITY

AUXILIARY

FUNCTIONS

(HOMING, LIMITS,

EMERGENCY STOP)

INVERSE KINEMATICS

TRANSFORMATION

C5E DRIVE

AXIS C

C5E DRIVE

AXIS B

C5E DRIVE

AXIS A

ETHERCAT REAL-TIME BUS

Figure 5: Block schematic of the experimental station’s

control software.

Machine used for testing purposes was all-

aluminum KOSSEL linear delta 3D printer. The Kos-

sel is a modular system for linear delta machines, that

focuses on upgradeability and easy manipulation of

workspace dimensions. This particular machine used

GT2 toothed belts driven directly by the servo-stepper

drives with linear guides (HIWIN MGN12) for in-

creased stiffness. The mechanical parameters of the

machine, are:

• arm length (L): 195mm

• arm radius (r): 65mm

• arm offset (Oxy): 30mm

The picture of the experimental station is shown in

Fig.6.

Feedrate Planning for a Delta Parallel Kinematics Numerically Controlled Machine using NURBS Toolpaths

747

Figure 6: Picture of the Kossel machine used for experi-

mental testing (top) with Nanotec C5E servo-stepper drives

(bottom).

6 EXPERIMENTAL RESULTS

The feedrate proﬁle was generated according to the

feedrate planning algorithm described in Section 4

for the NURBS toolpath presented in Fig.1. Kine-

matic limits were set to V

max

= 100mm/s,A

max

=

1000mm/s

2

,J

max

= 20000mm/s

3

. Maximum chord

error was set to ε = 0.001mm. Figure 7 presents the

generated feedrate proﬁle along with. cartesian axis

velocity, acceleration and jerk. The thick blue line on

the ﬁrst plot indicates the feedrate proﬁle. The dashed

lines indicate chord error, centripetal acceleration and

jerk components of the Feedrate Limit Function (or-

ange, green and violet respectively). The pink dots

indicate the identiﬁed critical points between which

the s-curve feedrate proﬁles were planned. The sub-

sequent subplots show velocity, acceleration and jerk

in each cartesian axis (blue - X, orange - Y, green -

Z). The dashed black lines indicate the positive and

negative limit of each value.

Fig.8 presents position, velocity, acceleration and

jerk of the linear joints. Fig.9 presents actual follow-

ing error of the linear joints.

It can be seen that the algorithm can limit the

cartesian velocity, acceleration and jerk to their lim-

its despite using the non-linear NURBS toolpath and

an S-Curve feedrate proﬁle. The actual joint lim-

its are not taken into account at this moment. Fur-

ther improvement of the feedrate generation will

lead to inclusion of linear joint limits by applying

Figure 7: Feedrate proﬁle with cartesian axis velocity, ac-

celeration and jerk.

Figure 8: Actual linear joint position, velocity, acceleration

and jerk recorded on the Delta Kossel machine.

1 2 3 4 5 6 7 8

Time[s]

0.15

0.2

0.25

0.3

Following error [mm]

Figure 9: Actual following error of the linear joints

recorded on the Delta Kossel machine.

the inverse kinematics transformation and optimiz-

ing the feedrate proﬁle parameters to meet the lim-

its.Furthermore additional constraints such as follow-

ing and contouring error will be added. The experi-

mental station will also be used for research of posi-

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

748

tioning error and torque ripple compensation meth-

ods using advanced controllers (Tarczewski et al.,

2014),(Tarczewski and Grzesiak, 2009).

In order to verify the computational efﬁciency of

the algorithm’s implementation in the real-time sys-

tem the feedrate generation process was repeated sev-

eral hundred times and the total computation time was

measured each time. The average computation time

of the whole proﬁle was 99.6ms with maximum and

minimum time equal to 104ms and 97.6ms respec-

tively. The total execution time of the toolpath from

Fig.1 with the feedrate proﬁle from Fig.7 was equal to

8.27s. This means that the proposed method has high

computational effectiveness. The computation time

of feedrate generation algorithm is much shorter than

the toolpath execution time. This means that the sys-

tem with the algorithm has on-line real-time capabili-

ties. Even with the inclusion of additional constraints

such as linear joint constraints the system should still

retain its real-time capabilities.

7 CONCLUSIONS

This paper presents a method for jerk limited fee-

drate planning for Non-Uniform Rational B-Spline

(NURBS) toolpaths in a parallel kinematics machine

in linear delta conﬁguration. The algorithm uses jerk-

limited feedrate planning with limitation of of carte-

sian axes velocity, acceleration and jerk. The pre-

sented experimental results show that the algorithm

can effectively constrain cartesian axis constraints.

Further improvements of the algorithm will constrain

velocity, acceleration and jerk in the linear joints. Fur-

thermore the computation times show that the algo-

rithm is computationally effective and is viable for

real-time implementation. Future research will in-

clude improvement of the algorithm and its extension

with additional constraints.

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