High accuracy spline interpolation for 5-axis machining

High accuracy spline interpolation for 5-axis machining

Matthias Mullera, Gabor Erdosb, Paul Xirouchakisa,*

aInstitute of Production and Robotics, EPF Lausanne CH-1015, SwitzerlandbComputer and Automation Research Institute, H-1518 Budapest, Kende u. 13-17, Hungary

Received 10 October 2003; received in revised form 7 January 2004; accepted 19 February 2004

Abstract

This article presents a new algorithm for simultaneous 5-axis spline interpolation. The algorithm basically merges two concepts: (1) the

interpolation of the toolpath with Pythagorean Hodograph (PH) curves and (2) the analytic solution of the inverse kinematic problem using

the template equation method. The first method allows one to obtain a analytic relation between the arclength and the parameter of the

toolpath curve. This one enables one to control the velocity of the tool on the workpiece. The second method allows one to determine the

analytic solution of the parameterized inverse kinematic problem that permits us to introduce arbitrary number of geometric parameters. A

natural selection of the possible parameters can be the parameters of tool geometry and the workpiece placement. This way, the off-line

generated inverse solutions—that transform the cutter contact curve into axis values—can be online compensated, as soon as the exact

parameter values are become known. Based on these two approaches, a robust and fast method for the simultaneous 5-axis spline

interpolation is developed. The result of this new algorithm is time-dependent axis splines which represent the given toolpath with high

accuracy.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Spline interpolation; Inverse kinematics; Template equation

1. Introduction

In this article a new spline interpolation algorithm for

simultaneous 5-axis machining is presented. Two concepts

are basically merged in the algorithm: (1) the representation

of the toolpath with PH curves and (2) the analytic solution

of the inverse kinematic problem obtained using the

template equation method.

Output of the proposed method is high accuracy and

time-dependent axis splines which results in a tool move-

ment with constant velocity in the cutter contact point. The

constant velocity is obtained by calculating the arclenght of

the cutter contact curve by using a 3D PH curve

interpolation. Using the desired constant feedrate, the

obtained arclength values can be transformed directly to

the corresponding time values, which are used as knot

sequence for the cubic spline interpolation on each axis.

To transform the cutter contact curve to the axis space of the

machine the analytic solutions of the inverse kinematic

problem is used. This transformation is calculated using the

template equation method, which allows one to use the

complete transformation between the cutter contact values

and the axis values containing the tool correction and the

workpiece placement as parameters. The fact that this

transformation given analytically is used to not only

calculate the axis values but also the derivative at each

axis values using the Jacobian matrix. This finally results in

high accuracy of the obtained axis splines.

2. Literature review

In this section the relevant research areas are reviewed

which relate to the 5-axis spline interpolation

problem. In the first part, the current state in the area

of NC spline interpolation is discussed. In the second part

an overview of the methods used to solve the inverse

kinematic problem of 5-axis machines using the template

equation method is given.

0010-4485/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.cad.2004.02.007

Computer-Aided Design 36 (2004) 1379–1393

www.elsevier.com/locate/cad

* Corresponding author. Tel.: þ41-021-693-2915; fax: þ41-021-693-

3553.

E-mail addresses: [email protected] (P. Xirouchakis), matthias.

[email protected] (M. Muller), [email protected]

(G. Erdos).

http://www.elsevier.com/locate/cad

2.1. PH curves

Real time interpolation is mainly concerned with the

control of the movements of the machine axes based on the

calculation of a sequence of points, that the tool must follow

at a given feedrate. Even in today’s NC controllers the

commonly employed interpolation technique is the linear

interpolation. This technique was introduced early by Koren

[20]. In his article he presented a software interpolator and

compared it to the hardware interpolators more often used at

this time.

In sculptured surface machining the toolpaths are in

general freeform curves. If the NC controller uses linear

interpolation these curved toolpaths have to be approxi-

mated by polylines. This process is generally refereed as

discretisation (see Rao and Sarma [29]). As stated by

Shpitalni et al. [21], and Kong and Yang [33] the linear

interpolation suffers from the following limitations, if it is

used for curved toolpaths: the discontinuities of the velocity

at the junctions of the linear segments result in high

acceleration, poor surface finish, low-surface accuracy and

long machining time. A lot of publications appeared in the

literature discussing the replacement of linear interpolation

techniques in NC control for sculptured surface machining.

One first approach to overcome the problem of linear

interpolation was based on replacing the linear segments

with circular arcs. A recent overview of this technique was

given by Li et al. [26]. This method applies only to 2D

curves.

As a more promising replacement for linear

interpolation, polynomial splines has been used to represent

curved toolpaths. Early publications like Morlec et al. [1],

Quan and Bedi [2,3], Shpitalni et al. [21] and Maekawa et al.

[22] used third order splines in Bezier form to represent

freeform toolpaths. Although this technique turns out to be

very powerful in representing any kind of smooth curve

shape, one major problem still remained: obtaining the

desired feedrate along the curve.

In the controller the interpolation points are generated

with a constant frequency, the sampling rate of the

controller loop. As a consequence, the time between two

consecutive points is constant during the interpolation.

Therefore, the feedrate has to be controlled depending on

the arclength distance between two consecutive points.

The calculation of the arclength distance is not

analytically solvable for general spline curves. The methods

developed to solve this problem can be divided into two

general approaches: (1) numerical calculation of the

arclength for a spline or (2) interpolation with special kind

of splines which have a relation between the arclength and

the curve parameter.

In the following publications the first approach is

discussed. Lo et al. [30] proposed a numerical method

based on a truncated Taylor series expansion to estimate

in real-time the next point along the spline for a given

feedrate. Yang and Chou applied the same approach for

3-axis [4] and for 5-axis machining [5,34] (Chou and

Yang) by adding analytical studies for the velocity,

acceleration and jerk control. Hsu and Yeh [35] applied

truncated Taylor expansion with an error minimisation

algorithm.

Yang and Wang [32], Wright and Wang [31] and Altintas

and Erkorkmaz [7] used the second approach to solve the

feedrate control problem. In their publications they

introduced splines with near-arclength parametrization. In

this technique the exact arclength parameterized spline is

approximated with a spline, which has at some points of the

spline the chord length as parameter. Although this

approach eliminates the drawback of the normal spline

interpolation, it implies still considerable off-line computing

effort. For example to generate the boundary conditions the

method uses a normal cubic spline interpolation as pre-

processing step. Another disadvantage of this approach is

that it uses the chord-length to parameterize the curve. This

method results only for curves with small curvature changes

in a near-arclength parameterization which is close to the

exact arclength parameterization.

Farouki [13,14] introduced the Pythagorean–Hodo-

graph (PH) curves to solve the problem of feedrate

control. This type of curves has the attractive property

that their arclength can be expressed as polynomial

function of the curve parameter. Farouki used these PH

curves to express toolpaths for sculptured surface

machining. He proposed in Ref. [12] a 2D Hermite

interpolation with Pythagorean–Hodograph (PH) and in a

series of articles [9–11,15,16] he discussed issues around

the use of PH curves for NC spline interpolation.

Farouki’s work in NC interpolation with PH curves

concentrated on 3-axis machining. This article extends

this work to NC spline interpolation algorithms for 5-axis

machines using 3D PH curves.

Spence and Fleisig [17] discussed the problem of spline

interpolation for 5-axis machining. They used the near-

arclength approach for the spline interpolation. In this

article PH-curves are used for the 5-axis spline interpolation

problem.

2.2. Template equation method

The inverse kinematics problem (in the following:

IKP) has been widely studied in robotics. From the

kinematic structure point of view the traditional industrial

robots (sometimes called manipulators) have open

kinematic chains. Since the kinematic structure of the

traditional 5-axis milling machines is very similar to the

manipulators, the same solution techniques can be used

in both cases.

The first solution of the IKP using template equations

was proposed by Paul [24,25]. He formulated the IKP of

some common industrial robot and also defined an

algorithm for the template-based solution. The template-

based solution technique has been used henceforth by

M. Muller et al. / Computer-Aided Design 36 (2004) 1379–13931380

many authors, who also introduced several new template

equations for different types of industrial robots [19,27,28].

By the evolution of the programming languages and

especially by the appearance of the symbolic manipulator

languages, new program packages have been developed,

that are capable of solving the IKP problems automatically

such as the ISG package of Halperin [18], the SRAST–

INKAS program of Herrera-Bendezu et al. [19], the SKIP

package of Rieseler, Schrake [27] and the Mathematica

based solver of Erdos [6]. In Ref. [23] the template

equation method was applied to obtain the transformation

presenting the inverse kinematics of general 5-axis

machines. This thesis shows how to obtain all the analytic

solutions of the inverse kinematic problem of 5-axis

machines.

One of the strengths of the template equation method

is its insensitivity on the number of parameters of the

IKP. This implies, that arbitrary number of parameters

can be introduced into the kinematic model of the 5-axis

machine, and the solution of the IKP can be automatically

obtained. Even if theoretically the IKP of 5-axis milling

machines are always solvable (this follows from the fact,

that the positioning and the orientation equations are

always decoupled), in practice usually hand solutions are

used, that are valid for a family of 5-axis milling

machine. To introduce new parameters into these

solutions is a tedious and error prone procedure. In this

article, the tool geometry and the workpiece placement of

the milling machine are parameterized, and the Mathe-

matica based solver developed by ErdHos is used to solve

the resulted IKP.

3. 5-Axis spline interpolation

In this section the 5-axis interpolation algorithm is

presented in detail.

3.1. 5-axis toolpath definition

For the calculation of the axis displacement based on a

kinematic model containing the tool geometry as parameter

(see Ref. [23]), a toolpath has to contain the following

information

�pccðtÞ cutter contact curve

�dðtÞ tool direction function

�nðtÞ surface normal function along the toolpath

ð1Þ

In general these three entities can be obtained as parametric

functions of a parameter t: This is always true if the

workpiece is described by parametric surfaces and the

toolpaths are generated with parameter curves on these

surfaces

S¼�fsðu;vÞ ¼

fsxðu;vÞ

fsyðu;vÞ

fszðu;vÞ

0BB@

1CCA parametric surface

u¼ guðtÞ; v¼ gvðtÞ parameter functions

�pccðtÞ ¼

�fsðguðtÞ; gvðtÞÞ toolpath on s along parameter curve

ð2Þ

The surface normal function can be obtained directly from

the parametric surface using the parameter functions

snðu;vÞ ¼du�fs £ dv

�fs

kdu�fs £ dv

�fsk

¼�fsnðu;vÞ general surface normal

with : dx�f ¼

d�f

dx

�nðtÞ ¼

�fsnðguðtÞ;gvðtÞÞ surface normal along the toolpath

ð3Þ

To get the tool direction the unit tangent vector tan along the

toolpath is needed. This vector is very often referred as feed

direction.

tanðtÞ ¼

d�pccðtÞ

dtd�pccðtÞ

dt

�� ð4Þ

The tool direction can be expressed as a function of the

normal and tangent vector and two angles: the roll angle ðuÞ

and the tilt angle ðvÞ; see Fig. 1. The roll angle specifies the

inclination of the tool in the feed direction (between surface

normal and tangent vector). The tilt angle specifies the

direction of the tool perpendicular to the feed direction.

The tool direction can be calculated as follows

�dðtÞ ¼ sin

p

22v

� sin

p

22 u

� �nðtÞ þ cos

p

22 u

� tanðtÞ

�

þ cosp

22v

� �nðtÞ£ tanðtÞ

k�nðtÞ£ tanðtÞk

ð5Þ

Using the Inverse kinematics solution with tool correction

the 5-axis toolpaths ð�pcc;

�n;

�dÞ described in Ref. [23] can be

transformed directly to curves on each axis by replacing

the driving parameters in the equations of the IKP with

Fig. 1. Tilt and roll angle.

M. Muller et al. / Computer-Aided Design 36 (2004) 1379–1393 1381

the parametric expressions for�pcc;

�n and

�d: This will yield

analytic, parametric expressions for the curves on each axis.

Two problems arise from that approach:

† The parametric expression for the curves on each axis are

very large and not suitable for online calculations.

† The parameter t of the expressions has no relation to the

length of the curve, so the feedrate of the tool on the

workpiece cannot be controlled with the general

parameter t:

In order to control the feedrate along the tool path, the

length of the curve has to be calculated with respect to

the general parameter t: This represents the main issue for

the general CNC interpolation. The goal of the conventional

CNC interpolation is to generate a sequence of points on

each axis at the motion controller loop closing frequency

(controller sampling frequency). The tool should have

thereby a constant feedrate at the cutter contact point ð�pðtÞÞ

in order to minimize cutting force variations, to avoid

chatter and to produce a smoother surface finish. The actual

state of CNC interpolation is discussed in the literature

review of this article. In this article a new way of CNC

interpolation for 5-axis toolpaths is proposed. The goal of

that method is the following:

Goal of the new CNC 5-axis interpolation. The goal is to

generate on each axis of the 5-axis machine a spline

function which represents the given toolpath with high

accuracy and which is parameterized exactly with the

arclength of the toolpath (cutter contact curve).

The trajectory representation as a spline function is

chosen to allow the online calculation of the interpolation

points and to obtain small volumes for the data

exchanged towards the 5-axis machine controller. If

these splines are given with the arclength s of the cutter

contact curve as parameter, the speed of the tool can be

controlled very easily. This result is new compared to the

actual state of CNC interpolation. In general the

arclength parameterized spline curves can be derived

with respect to s; which will allow to calculate the

velocities and accelerations on each axis for a given

feedrate on the toolpath. If these values are exceeding

the maximal velocities or maximal accelerations on any

axis, the feedrate can be directly adapted by changing the

parameterization of the spline curves.

In the following it will be discussed how to reach the goal

defined above. As discussed before it is possible in general

to obtain for all necessary, not constant input data of the

inverse kinematics (in general the driving parameters�p;

�n;

�d)

an analytic, parametric expression with respect to the

general parameter t: It will be considered first how to

transform these expressions to a parametric form with the

arclength s as the parameter.

3.2. Transformation to PH curves

Consider that the analytic expressions for�pccðtÞ;

�nðtÞ and

�dðtÞ are given with t as general parameter without any

meaning. The goal is to represent these data in a parametric

form with the parameter s: the arclength of�pcc (the cutter

contact curve). Assume that the parameter t has values from

0 to T : For this interval a very coarse discretisation is chosen

disðt0; t1;…; tnÞ

0 # ti # T ; i [ ½0; n� dis ¼ ðt0; t1;…; tnÞ [ ½0; T�

t0 ¼ 0; tn ¼ T ; t0 , t1 , · · · , tn

ð6Þ

Compared to a linear approximation the approximation dis

consists of a considerable less amount of points. For each of

these parameters the cutter contact point, the unit normal

vector and the unit tangent vector can be calculated

�p0 ¼

�pccðt0Þ;

�p1 ¼

�pccðt1Þ;…ð

�p0;

�p1;…;

�pnÞ;

�n0 ¼

�nðt0Þ;

�n1 ¼

�nðt1Þ;…ð

�n0;

�n1;…;

�nnÞ;

tan0 ¼ tanðt0Þ; tan1 ¼ tanðt1Þ;…ðtan0; tan1;…; tannÞ;

ð7Þ

Between each neighboring pair of cutter contact points�pi

and�piþ1 a PH Hermite interpolation using the unit tangent

vectors tani and taniþ1 can be realized. This yields for each

segment between�pi and

�piþ1 a quintic PH spline ^

�piðtÞ with

the corresponding arclength function siðtÞ (see Ref. [12]).

^�p0ðtÞ; ^

�p1ðtÞ;…; ^

�pn21ðtÞ;

s0ðtÞ; s1ðtÞ;…; sn21ðtÞ with : t [ ½0; 1Þ

ð8Þ

The new parameter t can be considered as a local parameter

which goes from 0 to 1 in the interval between�pi and

�piþ1:

With the piecewise arclength function the arclength for each

parameter ti can be calculated

si ¼ sið0Þ; and sn ¼ sn21ð1Þ ðs0; s1;…; snÞ ð9Þ

The following discretisation of the tool path can be

calculated

ðs0; s1;…; snÞ ð�p0;

�p1;…;

�pnÞ;

ð�n0;

�n1;…;

�nnÞ; ð

�d0;

�d1;…;

�dnÞ;

ð10Þ

The discretisation given in Eq. (10) corresponds to the exact

discretisation of a toolpath which was given with respect to

the arclength (natural parametrization: unit tangent vector,

see Eq. (4)). For each of these points the corresponding axis

values can be calculated using the equations of the IKP with

tool correction (see Ref. [23]). This will yield a series of

points on each axis. Knowing the arclength of the

corresponding cutter contact points, one could try to make

a cubic spline interpolation on each axis using the arclength

values as knot sequence. This approach suffers from two

problems: (1) for a cubic spline interpolation not enough

data are available (start and end tangent on each axis have


to be estimated); (2) the accuracy of the cubic spline is not

very high due to the coarse discretisation. To overcome

these problems the Jacobian matrix of the inverse

kinematics is used.

3.3. Jacobian matrix

In Ref. [23] the analytic equations for the inverse

kinematic problem containing the tool correction are

derived. In the following these equations are presented as

a function of the time-dependent input data (in general these

are the driving parameters)

q1 ¼ f1ðpx; py; pz; nx; ny; nz; dx; dy; dzÞ





ð11Þ

where q1; q2; q3 are the translational and q4; q5 are the

rotational axis values.

It is true in general that these analytic functions can be

partially derived with respect to the driving parameters

px; py; pz; nx; ny; nz; dx; dy; dz; which will result in the Jaco-

bian matrix for the inverse kinematics

MiJac ¼

df1dpx

df1

dpy

df1

dpz

df1dnx

df1

dny

df1dnz

df1

ddx

df1ddy

df1ddz

df2dpx

df2

dpy

df2

dpz

df2dnx

df2

dny

df2dnz

df2

ddx

df2ddy

df2ddz

df3dpx

df3

dpy

df3

dpz

df3dnx

df3

dny

df3dnz

df3

ddx

df3ddy

df3ddz

df4dpx

df4

dpy

df4

dpz

df4dnx

df4

dny

df4dnz

df4

ddx

df4ddy

df4ddz

df5dpx

df5

dpy

df5

dpz

df5dnx

df5

dny

df5dnz

df5

ddx

df5ddy

df5ddz

0BBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCA

ð12Þ

Eq. (11) can be considered as a multi-arameter vector

function. The derivation of this function is actually a linear

combination of partial derivatives for each parameter. This

yields the Jacobian matrix (see Eq. (12)). The Jacobian

matrix maps the derivative of the driving parameters to the

corresponding derivatives of the axis values

q10

q20

q30

q40

q50

0BBBBBBBBB@

1CCCCCCCCCA¼MiJac

�p0

�n0

�d0

0BB@

1CCA ð13Þ

Due to the fact that the inverse kinematics solution is given

analytically the Jacobian matrix can be calculated exactly.

This offers the possibility to not only calculate the axis

values q1 to q5 for the given discretisation but also the

derivatives of the axis values assuming the derivatives of the

driving parameters are known. This will open the opportu-

nity to make a spline interpolation on each axis based on

position and tangent data. The question is now how to get

the derivatives of the driving parameters. Due to the fact that

the spline on the axes should be given as a function of the

parameter s (arclength of cutter contact curve), the

derivative must also be expressed with respect to s: Section

3.4 discusses how to get the exact derivatives of the driving

parameters with respect to s:

3.4. Calculation of the derivatives

In previous sections the parametric expressions for the

driving parameters�pccðtÞ;

�nðtÞ and

�dðtÞ are generated (see

Eqs. (2), (3) and (5)). These expressions can be derived in

general with respect to t: Consider that the reparameteriza-

tion with respect to s is given

t ¼ tðsÞ ð14Þ

With this equation the function for�pccðtÞ can be repar-

ameterized

�pccðtðsÞÞ ¼

�pðsÞ ð15Þ

The derivative of this function with respect to t can be

obtained using the chain rule

d�pccðtðsÞÞ

ds¼

d�pccðtðsÞÞ

dt

dt

ds¼

d�pccðtÞ

dtdsdt

ð16Þ

The arclength function ðs ¼ sðtÞÞ can be considered as

known (see Section 4.7). This means that the exact

derivative of the cutter contact curve with respect to the

arclength s can be calculated using Eq. (16). Due to the fact

that the other driving parameters are also given as general

functions of t the same approach can be applied to calculate

the derivatives with respect to s

d�pcc

ds¼

d�pccðtÞ

dtdsdt

d�n

ds¼

d�nðtÞdtdsdt

d�d

ds¼

d�dðtÞdtdsdt

ð17Þ

These equations can be used to calculate the values of the

derivatives for the given discretisation dis:

d�pcc

ds

� i¼

d�pccðtÞ

dtdsi

dt

; t ¼ 0; t ¼ ti

d�pcc

ds

� n¼

d�pccðtÞ

dtdsn21

dt

; t ¼ 1; t ¼ T

ð18Þ


d�n

ds

� i¼

d�nðtÞdtdsi

dt

; t ¼ 0; t ¼ ti

d�n

ds

� n¼

d�nðtÞdt

dsn21

dt

; t ¼ 1; t ¼ T

ð19Þ

d�d

ds

� i¼

d�dðtÞdtdsi

dt

; t ¼ 0; t ¼ ti

d�d

ds

� n¼

d�dðtÞdt

dsn21

dt

; t ¼ 1; t ¼ T

ð20Þ

With these equations the following toolpath discretisation

can be calculated

ðs0; s1;…; snÞ; ð�p0;

�p1;…;

�pnÞ; ð

�p0

0;�p0

1;…;�p0

nÞ;

ð�n0;

�n1;…;

�nnÞ; ð

�n0

0;�n0

1;…;�n0

nÞ;

ð�d0;

�d1;…;

�dnÞ; ð

�d0

0;�d0

1;…;�d0

nÞ; with : f 0 ¼df

ds

ð21Þ

Using the inverse kinematics solution (Eq. (11)) and the

Jacobian of the inverse kinematics (Eq. (12)), the joint

values and their derivatives can be calculated for the given

discretisation with respect to the arclength s: The result will

be

ðs0; s1;…; snÞ; ð�q0;

�q1;…;

�qnÞ; ð

�q0

0;�q0

1;…;�q0

nÞ;

with : f 0 ¼df

ds;�q ¼ ðq1; q2; q3; q4; q5Þ

ð22Þ

3.5. Cubic spline interpolation

On the data derived in Section 4 a simple C1 piecewise

Cubic Hermite interpolation is applied (see Ref. [8]). This

interpolation yields a spline on each axis which represents

with reasonable accuracy the arclength parameterized joint

value function.

4. Example

4.1. Workpiece definition

In general free form surfaces are represented in

commercial CAD and CAM systems using parametric

surface patches. In this example the method presented in

Section 3.5 is applied to a simplified parametric surface

given by Eq. (23). Nevertheless the method will work in

the same way for the general case of a NURBS surface.

Sðu; vÞ ¼

200u

200v

150ð3 2 1:3ð20:5 þ uÞ2 þ 1:3ð20:5 þ vÞ3Þ

0BB@

1CCAð23Þ

For the interval u [ ½0; 1� and v [ ½0; 1� the surface S is

plotted in Fig. 2.

4.2. Discretisation of the toolpath

In this example the toolpath is discretised with 11 points

with a parameter-distance of 0.1 (see Eq. (6)).

disD0:1 ¼ {0:0; 0:1; 0:2;…; 0:9; 1:0} ð24Þ

4.3. Toolpath definition

On the surface S a toolpath is generated using the

following parameter function

uðtÞ

vðtÞ

!¼

t

t2

!ð25Þ

The expression for the cutter contact curve�pðtÞ is

�pðtÞ¼

200t

200t2

150ð321:3ð20:5þ tÞ2þ1:3ð20:5þ t2Þ3Þ

0BB@

1CCA ð26Þ

Fig. 2. Workpiece surface.


4.4. Normal vector function on the toolpath

The parametric equation for the surface normal along the

toolpath can be obtained from Eq. (3)

�nðtÞ¼

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4£109þtð26£10928£108tþ2£1010t321£1010t5ÞÞ

p

�

24£104þ8£104t

21£105ð20:5þt2Þ2

4£104

0BBBBB@

1CCCCCA ð27Þ

In Fig. 3 the surface normal vectors are plotted over the

toolpath. In this figure the unit surface normal vectors are

multiplied with a constant factor (50) to obtain a better

visual presentation.

4.5. Tangent vector

The parametric Eq. (28) for the tangent vector is

calculated based on Eq. (4). In Fig. 4 the unit tangent vectors

of the toolpath are plotted on the surface (also scaled for

visualization).

4.6. Tooldirection vector on the toolpath

The tool direction is calculated using Eq. (5). For this

example the following constant tool direction angles are

chosen

u ¼ 158 v ¼ 08 ð29Þ

4.7. Piecewise PH Hermite interpolation

Using the discretisation DD0:1 (see Eq. (24)) a set of cutter

contact points and of tangent vectors can be calculated using

Eqs. (26) and (28). These two data sets can be used to

calculate between each pair of neighboring discretisation

points a PH curve. The result will be a set of local PH curves

which can be used to generate a piecewise PH curve for the

entire toolpath which has C1 continuity, since the same

tangent vectors are used in the junction points for the

Hermite interpolation. In Fig. 5 the resulting piecewise PH

curve is shown.

In Fig. 5 it can be seen that the generated piecewise PH

curve seems to be identical in shape with the original toolpath

curve shown in Fig. 3. But it should be emphasized that the

piecewise PH curve is from a mathematical point of view

only an approximation of the original toolpath. In the

following the difference between the original toolpath and

the piecewise PH curve is analyzed quantitatively. The

original toolpath (Eq. (26)) and the piecewise PH curve are

general space curves. To compare both curves quantitatively

Fig. 3. Normal vectors.

tanðtÞ ¼

200ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi40; 000 þ 160; 000t2 þ 22; 500ð22:6ð20:5 þ tÞ þ 7:8tð20:5 þ t2Þ2Þ2

p400tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

40; 000 þ 160; 000t2 þ 22; 500ð22:6ð20:5 þ tÞ þ 7:8tð20:5 þ t2Þ2Þ2p

150ð22:6ð20:5 þ tÞ þ 7:8tð20:5 þ t2Þ2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi40; 000 þ 160; 000t2 þ 22; 500ð22:6ð20:5 þ tÞ þ 7:8tð20:5 þ t2Þ2Þ2

p

0BBBBBBBBBB@

1CCCCCCCCCCA

ð28Þ

Fig. 4. Tangent vectors.


the curves are projected to the x–z plane and to the y–z plane,

respectively, as shown in Fig. 6.

The projected curves are given, like the original curves,

also in parametric form. For the exact curve the parametric

form can be transformed analytically to the explicit form

(for the x–z plane z ¼ f ðxÞ or for the y–z plane z ¼ f ðyÞ).

The projections of the piecewise PH curve are

transformed numerically to explicit form using Newton

iteration for a given resolution. In each plane the z-values

of the projections of the piecewise PH curve are subtracted

from the z-values of the projections of the original

toolpath. This difference is referred in the following as z

Error: In Fig. 7 the z errors for both planes are plotted over x

and y:.

In the following the maximum error for the given

example is estimated. The calculated error in xðzErrorxÞ is

perpendicular to the error in yðzErroryÞ: The error in space

can be calculated with

Error ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffizError2

x þ zError2y

qð30Þ

If the maximum values for zErrorx and zErrory are used

to estimate the distance in space between the original

toolpath and the piecewise PH curve then in the worst

case a 0.026 mm maximum error results for an

interpolation length of about 23 mm (interpolation length

is the estimated average length of one interpolation

segment). This interpolation error seems to be reason-

able. The benefit of this interpolation is that the toolpath

is now represented with a piecewise PH curve where the

arclength can be calculated analytically from the

parameter.

The error introduced by the PH Hermite Spline

interpolation does not apply to the shape precision of the

trajectories, since the original cutter contact curve�pcc is

used to generate them. The PH Spline curve is only used for

the analytic arclength calculation and derivation. The

original curve and the PH Spline curve are very close (see

error discussion above). For this reason the error in the

arclength can be considered as negligible.

With Eq. (9) the arclength of each point of the

discretisation can be calculated, using the arclength function

of each segment obtained from the piecewise PH Hermite

interpolation.

ðs0;s1;…;s10Þ ¼ð0;27:67;55:01;82:14;109:24;137;

166:81;200:07;237:3;277:65;320:83Þ ð31Þ

4.8. Inverse kinematics

In previous sections the driving parameter set for

the inverse kinematics was calculated. As an example

the Ferrari A15 5-axis milling machine discussed in

Fig. 7. Error in z for (a) over x and for (b) over y:

Fig. 6. Projection of the toolpath to the x–z and y–z planes.

Fig. 5. Toolpath: piecewise PH curve.


Ref. [23] is used for the actual machining. In Fig. 8 the

position of the workpiece on the machine is illustrated.

In Fig. 9 the points on each axis are connected with line

segments (linear approximation). These figures represent a

linear interpolation of the axis values with respect to

the arclength of the toolpath. To evaluate the following

steps quantitatively a reference curve on each axis is

generated. This reference curve is based on the discretisa-

tion of 1000 points on the given toolpath. For all these points

the driving parameters and the corresponding arclength is

calculated. Then for each point the corresponding axis

values are calculated. If these axis values are plotted as a

polyline over the arclength values a quasi-exact represen-

tation of the axis values over the arclength is obtained. The

calculation of these reference curves took on the Mathe-

matica kernel about 10 min.

In Fig. 10 it is shown that the linear polyline based on the

normal discretisation is not sufficient for a precise machin-

ing of the workpiece surface.

In practical use the polyline on each axis is very often

smoothed by fitting a spline curve through the points of the

polyline. In general a cubic Spline interpolation is used to

realize a curve on each axis. In the following this approach

is tested and compared to the reference curve. For the cubic

Spline interpolation two things beside the interpolation

points are needed: (1) a knot sequence and (2) the start and

end tangent of the Spline curve. For the knot sequence the

already calculated arclength values are used. In practical

applications this should be calculated numerically. Here the

exact values obtained from the piecewise PH Spline Curve

are used. To get the start and the end tangent the reference

curves are used. For both cases, knot sequence and tangents,

the quality of the data values used for the evaluation of

Fig. 9. Linear approximation of the curves on each axis.

Fig. 8. Workpiece on Ferrari machine.


the cubic spline curves is higher compared to practical

applications.

In Fig. 11 the cubic spline curves obtained from the

discretisation points are compared with the reference curve.

The plotted error curves shown under the axis curves are the

difference between the reference curve and the cubic Spline

curve. If the maximal errors in these curves are normalized

with the corresponding axis value range a maximal relative

error of about 10% is obtained.

The derivatives of the driving parameters are calculated

using Eqs. (18)–(20) and the arclength functions obtained

from the piecewise PH Spline. In Fig. 12 the tangents are

plotted on the reference curve for the first axis.

On each axis the axis positions and their derivatives are

used for a C1 Piecewise cubic Hermite interpolation using

the arclengths as knot sequence (see Ref. [8]). This yields a

Spline curve on each axis with arclength parameterization.

In Fig. 13 these splines are compared to the reference

curve. It can be observed that these spline curves have an

error, which is on average approximately five times smaller

than the error of the cubic spline curves, which do not use

the tangent. Another point which could be observed is that

a relative large error amplitude occurs at an arclength of

about 123 mm. The value 123 mm is estimated by

analyzing the local maxima and minima on the error

curves. This relatively high error comes from the interplay

between the local curvature of the Spline curves and the

discretisation of the toolpath. In Fig. 14 this effect is shown

for q2. The local maximum on this curve happens to be at

an arclength value of 123 mm. This point is approximately

in the middle of a toolpath section (see discretisation points

in Fig. 14). For the high curvature in that section a cubic

Spline is not able to interpolate the original shape with

enough accuracy.

An interesting question is, where does the high curvature

at 123 mm comes from, considering that on almost all axes

maximal curvatures are occuring at that point? It can be

considered that at this point the toolpath passes near

a singular point of the machine. A study of singularities of

5-axis machines can be considered as a future research

topic, which can be based on the analytic solution of the IKP

derived in this article.

To overcome this problem an additional discretisation

point is introduced at an arclength of 123 mm. After

repeating all the necessary calculations for the new

discretisation a new set of cubic sline curves on each axis

is generated. This set and the corresponding errors are

shown in Fig. 15. Compared to the initial tangent based

Spline curves, it can be seen that in general the error is small

and that the disproportional high error amplitudes at

123 mm arclength could be smoothed.

This example shows that the accuracy of the new Spline

interpolation depends on the distribution of the discretisa-

tion points on the toolpath. A more detailed investigation of

that topic is outside the scope of this paper and will remain

open for future research work.

Fig. 10. Comparison between reference curve and normal discretisation.


5. Conclusion

In this article a new algortihm is presented for 5-axis

spline interpolation. This method offers a higher geometric

accuracy due to the fact that not only toolpath position are

translated to axis values but also derivatives of the toolpath.

The result obtained from this method is time-dependent axis

splines with a higher accuracy compared to conventional

spline interpolation methods. The adapting of the inter-

polation spline to technological factors like maximal

Fig. 11. Comparison between reference curve and cubic Spline interpolation based on normal discretisation.

Fig. 12. Derivatives for q1:


allowed velocities and accelerations on each axis or altering

material removal rates are not discussed in this article.

The algorithm turned out to be very robust if the

following assumptions are true: (1) the given surface is

smooth (no geometric discontinuities); (2) the position and

tangent values can be calculated on the given surface and (3)

a reasonable discretisation is given. Point (1) and (2) seems

to be trivial but it turned out in practical applications that

the surfaces obtained from commercial CAD or CAM

systems do not always correspond to what was aspected

from visualization. The problem of finding a optimal

discretisation was excluded from the scope of this article

although the issue is important for the robustness of the

algorithm in a practical application.

Fig. 13. Comparison between tangent-based cubic Splines and reference curves.


In the area of grinding of complex 3D surfaces the

proposed method contains a great potential for practical

applications. In the Swiss watch industry over 90% of the

grinding of middle and high valuable watches a still done

manually. In this area of application the grinding operations

can be described geometrically only be 3D curves. The

position accuracy of the grinding operation is crucial

although only the visual aspect of the final part is important.

This results from the fact, that only a very small amount of

material is removed during one grinding path. If the

interpolation spline curve is smooth but slightly deformedFig. 14. Interplay between Error and discretisation on q2:

Fig. 15. Comparison between improved tangent-based cubic Splines and reference curves.


compared to the real surface, the first grinding path will

not touch the pre-finished surface at each place. To have

a contact of the grinding tool at each place of the surface

several paths have to be done before the real grinding

process can start. Due to the small amount of

removed material this can result in a considerable

augmentation of the machining time, which is

definitely not desired. A preliminary implementation of

the method—discussed in this article—on a PC based

controller, showed to be very promising for the grinding

application. The implementation showed also that the

accuracy of the velocities on each axis is not crucial. To

obtain the desired result the velocity profiles should be only

smooth. Due to the fact that we use cubic spline curves on

each axis this condition is full filled.

Acknowledgements

This research was partially funded by the following

contracts between the Swiss Federal Office of Education and

Science and EPFL: the ESPRIT project 29708 STEP-NC

(STEP-Compliant Data Interface for Numeric Controls) and

the IMS STEP-NC project 01.222-1.This support is grate-

fully acknowledged.

References

[1] Andre P, Haddad MC, Morlec C. Application of uniform cubic

b-spline curves to machine-tool control. J Intell Robot Syst 1991;4.

[2] Bedi S, Ali I, Quan N. Advanced interpolation techniques for n.c.

machines. J Engng Ind 1993;115.

[3] Bedi S, Quan N. Spline interpolation technique for nc machines.

Comput Ind 1992;18.

[4] Chou JJ, Yang DCH. Command generation for three-axis cnc

machines. ASME J Engng Ind 1991;113.

[5] Chou JJ, Yang DCH. On the command generation of coordinated

motion of five-axis cnc/cmm machines. ASME J Engng Ind

1992;114.

[6] Gabor Erdos, A new symbolic approach for the geometric, kinematic

and dynamic modeling of industrial robots (in hungarian). PhD

Thesis, Technical University of Budapest; 2000.

[7] Erkorkmaz K, Altintas Y. High speed cnc system design. Part i: jerk

limited trajectory generation and quintic spline interpolation. Int J

Mach Tools Manuf 2001;41.

[8] Farin G. Curves and surfaces for CAGD. London: Acadamic Press;

1996.

[9] Farouki RT, Shah S. Real-time cnc interpolators for pythagorean

hodograph curves. Comput Aided Geom Des 1996;.

[10] Farouki RT, Manjunathaiah J, Nicholas D, Yuan G-F, Jee S.

Variable-feedrate cnc interpolators for constant material removal

rates along pythagorean–hodograph curves. Comput-Aided Des

1999;30(8).

[11] Farouki RT, Manjunathaiah J, Yuan G-F. G codes for the specification

of pythagorean–hodograph tool paths and associated feedrate

functions on open-architecture cnc machines. Int J Mach Tools

Manuf 1999;.

[12] Farouki RT, Neff CA. Hermite interpolation by pythagorean

hodograph quintics. Math Comput 1995;.

[13] Farouki RT, Sakkalis T. Pythagorean hodographs. IBM J Res Dev

1990;.

[14] Farouki RT, Sakkalis T. pythagorean–hodograph space curves. Adv

Comput Math 1994;.

[15] Farouki RT, Tsai YF, Wilson CS. Physical constrains on feedrates and

feed acceleration along curved tool paths. Comput Aided Geom Des

2000;17.

[16] Farouki RT, Tsai YF, Yuan G-F. Contour machining of free-form

surfaces with real-time ph curve cnc interpolators. Comput Aided

Geom Des 1999;.

[17] Fleisig RV, Spence AD. A constant feed and reduced angular

acceleration interpolation algorithm for multi-axis machining.

Comput-Aided Des 2001;33.

[18] Halperin D. Automatic kinematic modelling of robot manipulators

and symbolic generation of their inverse kinematics solution.

In: Stifter S, LenarcWicW J, editors. Advances in robot kinematics,

Wien: Springer; 1991. p. 310–7.

[19] Herrera-Bendezu LG, Mu E, Cain JT. Symbolic computation of robot

manipulator kinematics. Proc IEEE Int Conf Robot Automat 1988;2:

993–8.

[20] Koren Y. Interpolator for a computer numerical control system. IEEE

Trans Comput 1974;25(1).

[21] Koren Y, Lo CC, Shpitalni M. Cnc interpolators: algorithms and

analysis. Manuf Sci Engng 1993;64.

[22] Lartigue C, Thiebaut F, Maekawa T. Cnc tool path in terms of b-spline

curves. Comput-Aided Des 2001;33(3):xxxx.

[23] Matthias Muller, A new approach for trajoctory generation in 5-axis

machining. PhD thesis, Fcole Polytechnique Federal de Lausanne;

2001.

[24] Paul RP. Robot manipulators: mathematics programming and cont-

roll. Cambridge, MA: The MIT press; 1981.

[25] Paul RP, Shimano B, Mayer G. Kinematic control equations for

simple manipulators. IEEE Trans Syst, Man Cybernet 1981;11(6):

449–55.

[26] Qiu H, Cheng K, Li Y. Optimal circular arc interpolation for nc tool

path generation in curve contour manufacturing. Comput-Aided Des

1997;29(11).

[27] Riesler H, Schrake H, Wahl FM. Symbolic computation of closed

form solution with prototype equations. In: Stifter S, LenarcWicW J,

editors. Advances in robot kinematics. Wien: Springer; 1991.

p. 343–51.

[28] Riesler H, Wahl F. Fast symbolic computation of the inverse

kinematics of robots. In: Proceedings of IEEE International

Conference on Robotics and Automation, Cincinatti, vol. 1.; 1990.

p. 462–67.

[29] Sarma R, Rao A. Discretizors and interpolators for five-axis cnc

machines. Proceedings of the ASME Dynamic Systems and Control

Division, 64; 1998.

[30] Shpitalni M, Koren Y, Lo CC. Realtime curve interpolators. Comput-

Aided Des 1994;26(11):ccccc.

[31] Wang FC, Wright PK. Open architecture controllers for machine

tools, part 2: a real time quintic spline interpolator. J Manuf Sci Engng

1998;120.

[32] Wang FC, Yang DCH. Nearly arc-length parametrized quintic-spline

interpolation for precision machining. Comput-Aided Des 1993;

25(5).

[33] Yang CH, Kong T. Parametric interpolator versus linear

interpolator for precision cnc machining. Comput-Aided Des

1994;26(3).

[34] Yang DCH, Chou JJ. Automatic generation of piecewise constant

speed motion with smooth transition for multi-axis machines. J Mech

Des 1994;116.

[35] Yeh SS, Hsu PL. The speed-controlled interpolator for machining

parametric curves. Comput-Aided Des 1999;31.


Matthias Muller was born in Germany in

1974. In 1997, he recieved a diploma in

mechanical engineering from the Technical

University of Braunschweig, Germany. In

1992 he finished his PhD thesis at the

CAD/CAM Laboratory of the Swiss Fed-

eral Institute of Technology at Lausanne,

Switzerland. Currently he is working as

project manager at the SmartKinematics

Company. In his current position he is

responsible for software development pro-

jects is the area of CAD/CAM.

Professor Paul Xirouchakis is directing

the computer-aided design and computer-

aided manufacturing (CAD/CAM) labora-

tory, institute of production and robotics at

the Swiss Federal Institute of Technology

in Lausanne, Switzerland. His research

interests are in the areas of (i) product

modeling and reasoning for manufacture/

assembly/remanufacture (ii) manufacturing

information systems and (iii) informatics

for planning and scheduling for manufac-

ture/assembly/remanufacture. Current project activities include: CAD

tools for conceptual design of mechanical assemblies, geometrically

intelligent NC-controllers built on feature-based post-processors and

planning and scheduling of remanufacturing systems. Professor Xirou-

chakis has industrial experience with CAD industries in the USA where he

directed computer-aided simulation-based design projects. He has been a

professor at the National Technical University of Athens and the

Massachusetts Institute of Technology in the area of Structural

Mechanics. He obtained his Ph.D. in Structural Mechanics in 1978

from Massachusetts Institute of Technology.

Gabor Erdos is a Senior Research

Associate at the Computer and Automation

Research Institute. He received his MS and

PhD degrees in mechanical engineering

from Budapest University of Technology

and Economics. He also holds an MS

degree in mechanical and aerospace engin-

eering from SUNY Buffalo. Dr Gabor

Erdos was a post-doc assistant for five

years at Institute of Production and

Robotics at EPF Lausanne. His main

research interests include computer aided mechanism modeling, data

exchange between CAD-CAM-CNC and machining simulation.


Documents

High accuracy spline interpolation for 5-axis machining