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Doctoral Thesis

Exploitation of tolerances and quasi-redundancy for set pointgeneration

Author(s): Sellmann, Florian

Publication Date: 2014

Permanent Link: https://doi.org/10.3929/ethz-a-010335533

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Diss. ETH No. 21975

Exploitation of tolerances andquasi-redundancy for set point

generation

A dissertation submitted to the

ETH ZURICH

for the degree of

Dr. sc. ETH Zurich

presented by

FLORIAN SELLMANN

Dipl.-Ing. Maschinenbau, TU Darmstadt

born 4. April 1981

citizen of Germany

accepted on the recommendation of

Prof. Dr. K. Wegener, examiner

Prof. Dr. C. Glocker, co-examiner

2014

Acknowledgments

This thesis was realised during my time at inspire AG and the Institute for Machine Tools

and Manufacturing (IWF) of the ETH Zurich.

The cooperation with my co-workers and with the industrial project partners greatly

contributed to my work. I would like to thank Prof. Dr. Konrad Wegener, head of the

IWF and supervisor of this thesis for his generous support, constructive suggestions and

for his trust in me and my work.

My group leader Dr. Sascha Weikert and my co-supervisor Prof. Dr.-Ing. Dr.-Ing. ha-

bil. Christoph Glocker supported my research and thesis and contributed with valuable

suggestions and ideas.

This work would not have been the same without the support of my industrial partners.

In particular, I am indebted to Dr. Gerhard Hammann, Dr. Nico Zimmert and Thomas

Kieweler from TRUMPF as well as Dr. Martin Munz and Ralf Spielmann from the Siemens

AG.

I appreciate the supportive teamwork of Daniel Spescha, Titus Haas, Dr. Markus Steinlin,

Hop Nguyen, and Stefan Thoma. Thank you for the inspiring discussions and critical

remarks. Additionally I would like to thank all the people at and around the IWF, who

provided a comfortable working atmosphere for all the work- and non-work related dis-

cussions, especially Dr. Carl Wyen, Dr. Thomas Lorenzer, Dr. Josef Mayr and Florentina

Pavlicek.

In particular, I am deeply grateful to Dr. Florian Bachmann, who is not only a companion

but also a good friend and always supported me with valuable advices in all situations.

Last but not least I want to thank my family. My parents, who gave me a good education.

My sisters for their support and encouragement. And my wonderful wife Anna for all her

patience and support.

Florian Sellmann

Zurich, Dezember 2014

III

Contents

Symbols and abbreviations VIII

Abstract XII

Zusammenfassung XIII

1 Introduction 1

1.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 State of the art 5

2.1 Set point generation for machine tools . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Fundamental ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Geometry optimisation for machine tools . . . . . . . . . . . . . . . 7

2.1.3 Feed rate optimisation for machine tools . . . . . . . . . . . . . . . 12

2.2 Feed rate optimisation for redundant machine tools . . . . . . . . . . . . . 14

2.3 Error modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Research gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Fundamentals of curves 19

3.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Important properties of curves . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Orientation representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Methods for interpolation and approximation . . . . . . . . . . . . . . . . 24

IV

3.4.1 Polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.2 B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.3 NURBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Path dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 Continuity of trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Geometry optimisation for machine tools 31

4.1 Global optimisation problem . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Local optimisation problem . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.1 Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.3 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Discrete geometry optimisation . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.1 Quadratic programming problem . . . . . . . . . . . . . . . . . . . 38


4.3.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.4 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3.5 Parameterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.6 Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Application examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4.1 Virtual CNC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4.2 Machine model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4.3 2D laser cutting example . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4.4 3D laser cutting example . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Feed rate optimisation using quasi-redundant degrees of freedom 64

5.1 Definition of quasi-redundancy . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Conception of the optimisation . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3.1 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

V

5.3.2 Geometric Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3.3 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.4 Nonlinear programing approach . . . . . . . . . . . . . . . . . . . . . . . . 71


5.4.2 Numerical optimisation . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4.3 Application example . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5 Quadratic programing approach . . . . . . . . . . . . . . . . . . . . . . . . 76

5.5.1 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.5.2 Dynamic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 80


5.5.4 Influence of the additional weights . . . . . . . . . . . . . . . . . . . 81

5.6 Application Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Tolerance model 92

6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2.2 Calculation of geometric uncertainties . . . . . . . . . . . . . . . . . 95

6.2.3 Overall description of the method . . . . . . . . . . . . . . . . . . . 97

6.3 Geometric uncertainties of a 3-axis machine tool . . . . . . . . . . . . . . . 99

6.3.1 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4 Geometric uncertainties of a 5-Axis machine tool . . . . . . . . . . . . . . 101

6.4.1 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7 Conclusion and Outlook 108

7.1 Geometry optimisation for machine tools . . . . . . . . . . . . . . . . . . . 108

7.2 Feed rate optimisation using quasi-redundant degrees of freedom . . . . . . 109

VI

Appendix 111

A Parametric study 112

Bibliography 115

VII

Symbols, abbreviations and glossary

Symbols

a+ upper limit of rectangular distribution

a− lower limit of rectangular distribution

a Control points

a, b Interval limits

A Linear inequality constraint function: Ax ≤ b

Aeq

Linear equality constraint function: Aeqx = beq

b Linear inequality constraint function: Ax ≤ b

beq Linear equality constraint function: Aeqx = beq

C(i) Matrix with the function values of the ith parametric derivative of

a B-Spline

Cn nth time parametric continuity of a curve respectively continuity ofdnsdtn throughout the curve.

cv coverage factor

d Dimension of a parametric curve

et(s) Tangential accompanying vector

en(s) Normal accompanying vector

eb(s) Binormal accompanying vector

ex Unit vector in direction x

Gn nth time geometric continuity

h Number of NC-Blocks

I Identity matrix

i, j General index and counter variables

J Objective function of an optimisation problem

k Multiplicity of a knot

VIII

L Chord length of a spatial curve

m Number of control points

n Number of discrete points

n(s) Normal vector of the sheet metal

N B-Spline basis function

p Degree of a polynomial function

r(s) Path curve

R Rational Base Function

s Path parameter, also called curve parameter

t Time variable

u Knot vector

uc Combined standard uncertainty

ur Strongly positive correlated contributors for the combined standard

uncertainty

ui Standard uncertainty of uncorrelated contributors for the combined

standard uncertainty

U Measurement uncertainty

w Weights for NURBS shaping

x(i) ith derivative of x with respect to t respectively s

x∗ Optimal solution

δx Small change of x

∆ Euclidian length

∆n Tolerance for the deviation in direction of ev(s)∆t Tolerance for the deviation in direction of eu(s)∆proc Tolerance for the deviation in direction of ew(s)∆ϕtilt Tolerance for the deviation of the tilt-angle of the tool orientation

∆ϕlead Tolerance for the deviation of the lead-angle of the tool orientation

∆dwg Tolerance value known from the drawing

∆NC Tolerance value given to a machine tool without any machining error

∆smt Available smoothing tolerance

δϕ Small angular deviation

ζ Set of equality constraints for the description of the state equations

ηi Additional weights for the discrete geometry optimisation

θ(s) Local Frenet frame

κ Curvature of a spatial curve

λi Parameter for the shaping of B-Spline of degree while ensuring G2

continuity

IX

νi Additional weights for the exploitation of quasi-redundancy

Φ Physical limitation

ϕ Orientation angle

ψ(s) Local coordinate system for the tolerance sharing for linear axes

consisting of the eu(s), ev(s) and ew(s)0 Zero vector

Abbreviations

CAD Computer Aided Design

CAM Computer Aided Manufacturing

CNC Computerised Numerical Control

DAO Domain of admissible orientations

DGO Discrete geometry optimisation

DOF Degrees of freedom

ETH Eidgenossische Technische Hochschule Zurich

MCS Machine coordinate system

NC Numerical Control

NLP Nonlinear programming

NURBS Non Uniform Rational B-Splines

QP Quadratic programming

TCP Tool Center Point

Glossary

Closed loop control Control with feedback of the actual position.

CNC/NC Abbr.: Computerised Numerical Control. Automatic control of a

process performed by a device that makes use of numerical data

introduced while the operation is in progress according to [23].

Feed rate Process speed programmed in the NC-program.

Feed rate optimisa-

tion

Sub step to the trajectory generation, where the feed rate is opti-

mised.

Geometry optimisa-

tion

Sub step to the trajectory generation, where only the geometry is

optimised.

Jerk Third time derivative of position d3xdt3 .

X

Open loop control Prediction of the feed rate without any feedback of the actual posi-

tion.

Semi closed loop con-

trol

Closed loop control with an indirect measurement system.

Set point Output of the trajectory generation. Desired value of the actuator.

Tracking error Difference between the actual position of an axis and the previously

calculated set points.

Trajectory Toolpath as a function of time.

Trajectory generation The path and feed rate provided by the NC program is optimised

by the trajectory generation. The resulting trajectory satisfies the

demands to the trajectory generation algorithm.

XI

Abstract Abstract

Abstract

An increase in productivity of a machine tool is obtained by reducing the processing time,

while maintaining the required quality within given constraints, which result from the

features of the machine structure, the actuators and the process. The set point gener-

ation, consisting of geometry optimisation and feed rate optimisation, is minimising the

processing time within machine specific limits.

This thesis presents an algorithm for discrete geometry optimisation using B-splines, and

allows for a pointwise tolerance for deviations from the original geometry for any number

of axes. By the use of a tolerance model the geometric uncertainties of the machine

tool can be considered for the pointwise evaluation of the smoothing tolerance. For the

subsequent feed rate optimisation higher parametric derivatives are optimised during the

optimisation. The discretised geometry is optimised by a quadratic programing approach.

Depending on the manufacturing process the manufacturing tolerances are shared by the

different axes, which is demonstrated at the example of a 5-axis laser cutting machine

tool. Application examples show, that the geometry optimisation leads to an increase of

the machining productivity over state of the art methods.

Furthermore a method for the exploitation of the quasi-redundancy is presented in this

thesis based on additional degrees of freedom, which arise in 5-axis machining depending

on the manufacturing process. Taking into account the inertias of the different axes the

exploitation of the quasi-redundancy is demonstrated at the example of a 5-axis laser

cutting machine tool with two quasi-redundant degrees of freedom. The optimisation of the

weighted jerks of the different axes in response to the quasi-redundant degrees of freedom

results in a nonlinear optimisation problem for the general case of a large orientation

tolerance with either linear or nonlinear constraints. For the use case of a small orientation

tolerance a linearisation of the optimisation problem mentioned above is obtained, which

leads to a quadratic programming problem. Thus for a given trajectory at the Tool-Center-

Point a jerk minimal trajectory of all axes is determined. Application examples show, that

the mechanical excitation of the machine tool and the path disturbance can be reduced.

XII

Zusammenfassung Zusammenfassung

Zusammenfassung

Eine Produktivitatssteigerung einer Werkzeugmaschine ergibt sich durch Reduktion der

Bearbeitungsdauer unter Beibehaltung der geforderten Qualitat innerhalb gegebener Rand-

bedingungen. Diese ergeben sich durch die Eigenschaften der Maschinenstruktur, der An-

triebe und des Prozesses.

Die Fuhrungsgrossengenerierung, bestehend aus Geometrieoptimierung und Geschwindig-

keitsfuhrung, minimiert die Bearbeitungsdauer innerhalb maschinenspezifischer Limite.

Diese Arbeit prasentiert einen neuen Algorithmus zur diskreten Geometrieoptimierung

mittels B-Splines beliebiger Ordnung, welcher fur beliebig viele Achsen eine punktwei-

se Tolerierung der Abweichungen von der Originalgeometrie berucksichtigt. Mittels eines

Toleranzmodells konnen fur jedem Punkt im Arbeitsraum geometrische Unsicherheiten

bei der zur Verfugung stehenden Toleranz berucksichtigt werden. Fur die anschliessende

Fuhrungsgrossengenerierung werden wahrend des Optimierungsschritts hohere parametri-

sche Ableitungen optimiert. Im Algorithmus erfolgt eine Formulierung als ein quadrati-

sches Optimierungsproblem, welches mit einem Standardverfahren gelost werden kann.

In Abhangigkeit des Fertigungsverfahrens erfolgt eine Aufteilung der Fertigungstoleran-

zen auf die einzelnen Achsen, was am Beispiel des 5-achsigen Laserschneidens fur Linear-

und Rundachsen dargelegt wird. Anhand von Anwendungsbeispielen wird aufgezeigt, dass

sich durch die beschriebene Geometrieoptimierung eine Produktivitatssteigerung gegen-

uber herkommlichen Verfahren ergibt.

Des Weiteren wird in dieser Arbeit ein Verfahren zur Ausnutzung der Quasi-Achsredundanz

prasentiert. Hierbei handelt sich um zusatzliche Freiheitsgrade bei der 5-Achs-Bearbeitung,

welche sich in Abhangigkeit des Fertigungsverfahrens ergeben. Die Auflosung der Quasi-

Achsredundanz wird am Beispiel des 5-achsigen Laserschneidens mit zwei quasi-redundanten

Freiheitsgraden und unter Berucksichtigung der Achstragheiten aufgezeigt. Durch Optimie-

rung der gewichteten Achsrucke in Abhangigkeit der inneren Freiheitsgrade ergibt sich fur

den allgemeinen Fall grosser Orientierungstoleranzen ein nichtlineares Optimierungspro-

blem mit wahlweise linearen- oder nichtlinearen Nebenbedingungen. Fur den Anwendungs-

XIII

Zusammenfassung Zusammenfassung

fall mit kleiner Orientierungstoleranz wird eine Linearisierung des o.g. Optimierungspro-

blems durchgefuhrt. Hieraus ergibt sich ein quadratisches Optimierungsproblem, welches

sich mit einem Standardverfahren losen lasst. Somit lasst sich zu einer gegebenen Tra-

jektorie am Tool-Center-Point eine ruckminimale Trajektorie aller Maschinenachsen be-

stimmen. Anhand von Anwendungsbeispielen zeigt sich, dass sich der Ruck der einzelnen

Maschinenachsen, die mechanische Anregung der Maschine und somit Bahnabweichungen

reduzieren lassen.

XIV

1

Chapter 1

Introduction

This thesis describes methods for the set point generation of machine tools for high produc-

tivity applications. The aim of these machines is to produce workpieces within a specified

accuracy and a minimum amount of machining time.

From the design of a workpiece to the final manufacturing process there are many steps

in between,which have to be passed as shown in figure 1.1.

Figure 1.1: Manufacturing data flow.

A common tool for the design of a workpiece is the computer aided design (CAD). Cur-

rent CAD programs offer various capabilities for the design of a workpiece, the virtual

assembling of different components, and also FEM simulations of the workpiece. The de-

2 1. Introduction

sign informations of the CAD include the geometry of the workpiece as well as tolerance

informations.

Due to the fact that the final manufacturing process consists of a relative movement of

the tool and the workpiece along spatial trajectories the surface informations of the design

have to be converted into NC code to be used for the control of the machine tool. This

step is performed by a computer aided manufacturing (CAM) tool like NX [70].

The resulting NC code contains the segment wise geometric informations of the trajecto-

ries, a proposed feed rate along the trajectories during manufacturing as well as specific

commands for the control of the process. Usually the geometric information of the trajecto-

ries is given to the machine tool by standard geometric elements like straight lines, circular

arcs, polynomial functions or B-Splines referred to ISO 6983-1 [48] and ISO66025-2 [45].

Based on the NC-Code the control manages the movement of the machine axes, which

interact with the mechanical structure of the machine tool as well as the process. This

interaction is observed by the measurement system assembled to the machine axes and

given to the control, which reacts to the interaction.

The procedure inside of the computerised numerical control (CNC/NC) can be divided into

open loop and closed loop controller according to Stute [81]. In the open loop controller

the set points for the movement of the machine axes are calculated according to figure 1.1.

Because of the required real time execution of NC code, the set point generation of state

of the art machine tools is divided in the two main steps geometry optimisation and

feed rate optimisation. In a first step the geometry, which is given by the NC-Code

is prepared, since discontinuities are eliminated. The subsequent feed rate optimisation

generates a time discrete feed rate profile along the previously optimised geometry fulfilling

the constraints, which are given by the limits of the machine tool and additional commands

like the proposed feed rate along the tool path e.g. included in the NC code. Afterwards,

the fine interpolated trajectory is given to the closed-loop controller of the different axes,

which finally leads to a relative motion between the tool and the workpiece. During

manufacturing the closed loop controller adjusts the deviation between the precalculated

set points and the actual movement of the machine tool. For further particulars it is

referred to Geering [32] and Stute [81].

An important parameter for both the manufacturers and the users of machine tools is

the productivity, which is defined as the number of manufactured parts of a comparable

quality per manufacturing time according to Weck et al. [90] and is influenced by both the

accuracy and the machining time.

Both geometry and feed rate optimisation are having an impact on the productivity. Ac-

1.1 Outline of the thesis 3

cording to Hadorn [38], Yu [94] and Fauser [27] discontinuous movements and high curva-

tures usually lead to mechanical excitations and a reduced accuracy of the machine tool

due to discontinuous and high actuator forces. Pursuant to Beudaert et al. [10] the con-

secutively optimised feed rate is influenced by the parametric derivatives of the previously

optimised geometry. The feed rate optimisation minimises the machining time by taking

advantage of the limits of the axes so the optimisation of the set points leads to

• Reduced mechanical excitation

• Exploitation of the axes limits

and finally an increased productivity.

Regarding the geometry optimisation a sufficient tolerance domain must be defined in a

previous step. Taking into account that the resulting deviation of a machine tool consists

of different contributors due to static, dynamic, thermal and process errors this thesis

presents an approach for the position dependent calculation of the smoothing tolerance

based on the known geometric errors of a machine tool.

A possible approach for a reduction of the mechanical excitation respectively high actuator

forces is a rearrangement of the forces, which are given to the mechanical system. Five

axis machine tools consisting of both linear and rotational axes offer the capability of

additional degrees of freedom within the orientation tolerance for the rearrangement of

the actuator forces with respect to the axes inertias. This work provides an optimisation

algorithm for the reduction of the mechanical excitation by rearranged actuator forces.

1.1 Outline of the thesis

The thesis is structured as follows:

After a brief introduction here in chapter 1, chapter 2 gives an overview of known methods

in the fields of set point generation and error modeling. Section 2.1 discusses the demands

on geometry optimisation by a brief introduction of the operative connection and presents

state of the art approaches for the geometry optimisation. Common approaches for the

resolution of redundancy are illustrated in section 2.2. State of the art approaches in the

field of error modeling are mentioned in section 2.3. The last section 2.4 specifies the

research gap, which is approached within this thesis.

The fundamentals of the curves used in this thesis are described in chapter 3. In sec-

tion 3.1- 3.3 general properties of the regarded curves including the representation of the

4 1. Introduction

orientation are presented. Section 3.4 illustrates known methods for the interpolation and

approximation of given data points.

A quadratic programming approach for both the rounding of the transition of subsequent

NC-Blocks and the compression of multiple NC-Blocks within a given tolerance is discussed

and quantified in chapter 4.

Chapter 5 presents two approaches for the exploitation of quasi-redundancy. In the case

of large tolerances of the tool orientation a nonlinear optimisation approach is presented

in section 5.4. For the use case of small tolerances of the tool orientation a quadratic

programming approach is shown in section 5.5.

In order not to suffer from the heuristic approach of assuming a global conservative toler-

ance value along the whole tool path, a procedure for the position dependent elaboration

of the available smoothing tolerances based on the geometric errors of a machine tool is

presented in chapter 6.

A summary of the results and an outlook of further research topics is presented in chapter 7.

5

Chapter 2

State of the art

This chapter gives an overview of known approaches in the field of set point generation

for machine tools. Both industrial and scientific approaches for the set point generation of

machine tools are presented in section 2.1. First of all fundamental ideas concerning the set

point generation are presented in section 2.1.1. After that methods for both geometry and

feed rate optimisation are illustrated in section 2.1.2 and section 2.1.3 respectively. Current

approaches for the exploitation of redundancy are regarded in section 2.2. Section 2.3

discusses several approaches in the field of error modelling for machine tools. Finally

section 2.4 identifies the research gap, which will be filled in this thesis.

2.1 Set point generation for machine tools

The requirement for a machine tool of maximum productivity is comparable to the task a

race driver has to accomplish. According to Braghin et al. [15] the best race driver is the

one that, with a given vehicle, is able to drive on a given track with a given track width

in the shortest possible time. Two strategies can be followed to reach this task according

to figure 2.1: minimise the space and/or maximise the speed respectively minimise the

curvature.

Knowing, that these two strategies are conflicting, the shortest lap time is reached by a

compromise between the two strategies. In this comparison the vehicle stands for the tool

respectively the machine tool, the track stands for the programmed tool path and the

track width finally stands for the given manufacturing tolerance.

Because of the requested real time ability of state of the art NC controllers the optimisation

of geometry and feed rate are subsequent calculation steps in order to save calculation time.

This is called a decoupled approach and is based on the heuristics, that an optimisation of

6 2. State of the art

a

b

Figure 2.1: Comparison between the shortest track (a) and the lowest curvature (b),

which allows for an increased speed along the track.

the tool path and a subsequent optimisation of the feed rate does not satisfy the demand

for a time optimal movement in general. However from a practical point of view in most

of the use cases the heuristic mentioned above is a sufficient trade off between robustness

respectively real time ability and the productivity.

2.1.1 Fundamental ideas

Non regarding point to point movements, which is a usual task for robot applications based

on the heuristic described above an usual task for a machine tool is the geometry optimi-

sation along curved tool paths. It contains all methods for the calculation of geometries

from a startpoint to an endpoint with a predefined shape within given tolerances. Due to

the fact that most of the geometric segments, which are used by state of the art CAM tools

are discontinuous (non tangential) straight lines and circular arcs, two main challenges for

the geometry optimisation along curved tool paths are the rounding of block transitions

and the comprehensive rounding of multiple NC-Blocks or in general an elimination of

discontinuities.

The subsequent feed rate generation maximises the feed rate along the previously defined

tool path in order to minimise the manufacturing time with respect to given limitations

due to the requested accuracy.

Independent from the used method for the set point generation in every point of the

resulting time dependent trajectory the interaction between the manufacturing tolerance,

2.1 Set point generation for machine tools 7

the feedrate along the tool path, the acceleration perpendicular to the tool path and the

curvature of the tool path, which is shown in figure 2.2 have to be considered.

acceleration

velocity

curvaturemanufacturing

tolerance

Figure 2.2: Operative connection for the set point generation

One of the influencing factors for the fulfillment of the manufacturing tolerances are the

dynamic errors of the machine tool. Depending on the compliance of the mechanical

structure these errors are affected by all acceleration derivatives perpendicular to the tool

path. These accelerations on the other hand depend on the feed rate along the tool path

and the curvature of the tool path itself. For the simplification of these interactions a

heuristic approach is the limitation of different dates of the machine tool. Not only for

limitation of dynamic errors but also for the stability of the manufacturing process itself a

usual way to achieve the demanded accuracy is the limitation of the feed rate. Further the

axis-wise acceleration and the jerk are usually limited in order to reduce the mechanical

excitation and to guarantee a certain accuracy at every point of the workspace.

Depending on the application a selection of both the commercial- and the scientific ap-

proaches for geometry optimisation are regarded below.

2.1.2 Geometry optimisation for machine tools

In the following both scientific and commercial state of the art approaches for geometry

optimisation in the fields of

• General smoothing methods

• Surface machining


• Pocket machining

• Collision avoidance

are presented and discussed.

General smoothing methods: Methods for geometry optimisation of machine tools,

which are independent from the manufacturing process focused on tool path smoothing as

well as 5-axis machining are summarised and discussed below.

For the optimisation of the geometry and the feed rate for linear systems Yu [94] presents

algorithms, which are based on the spline representation of the tool path and use several

spline based interpolation and approximation methods. Bouard et al. [12] minimises the

sum of the square of the curvature while ensuring the constraints given by the process and

the geometry. Zhao et al. [97] shows the analytic smoothing of multiple G01 blocks with

B-Splines. In addition feed rate constraints are evaluated along the tool-path.

The compression of multiple NC-Blocks includes all methods for smoothing of a geo-

metry by the replacement with a primarily continuous curve within the requested toler-

ances. Hadorn [38] investigated two smoothing algorithms: Transition smoothing with

quintic Bezier-Splines and comprehensive smoothing of multiple NC-Blocks with cubic

Beta-Splines. The curvature minimal shaping of the splines is done via an optimisation

algorithm. During optimisation for the fulfillment of the given tolerances the convex hull

property of the used Bezier respectively Beta-Splines is used, which leads to a resulting

continuous curve on the convex side of the original geometry but excludes all solutions

on the concave side of the original geometry according to figure 3.5. Established in the

field of CAGD Park [63] evaluates dominant points of a given discrete curve depending

on the curvature. As a consequence of that only discrete points at curve sections with

a large curvature are considered for a subsequent approximation with B-Splines. For the

approximation of curves with a continuous curvature this approach leads to a good ap-

proximation with a minimum amount of control points. In the case of discontinuities or

high curvatures of the geometry this approach leads to an unintended accumulation of

control points and thus a bad shape of the resulting curve as well as a low feed rate.

A quadratic programming approach with linear constraints for the smoothing of B-Splines

is formulated by Kano et al. [50]. The integral of the squared second derivative of a B-

Spline within the constraints is minimised here and exemplified. Although the deviation

between the smoothing function and the programmed trajectory is limited during opti-

misation, there is no tolerance sharing between the optimised axes, which are involved in

the optimisation. As a consequence of that a maximum utilisation of the manufacturing


tolerance is not given with this approach. Also Haas [37] uses a quadratic programming

approach for the optimization of the tool-path of a machine tool. In his thesis higher

parametric derivatives of the curve are minimised with respect to linear constraints on the

tool path. The decreased mechanical excitation is evaluated by the use of a 2 mass spring

model and measurements on a real machine tool.

5-axis machining has become more important in the last decade, because of an envisaged

decreased manufacturing time, enhancement of the quality of the workpiece, better cutting

conditions for the tool as well as the minimisation of subsequent manufacturing tasks,

which finally leads to a cost reduction. Taking into account that the feed rate is directly

influenced by the parametric derivatives of the geometry Beudaert et al. [10] show an

approach for the smoothing of 5-axis tool paths in order to maximise the resulting real

feed rate and to reduce the machining time by local smoothing of single axes movements.

In the case of geometrical variations the feed rate has to be decreased, so by a local

smoothing of the tool paths the real feed rate is maximised. So knowing the dynamic

limitations of each drive a maximum reachable feed rate is computed in order to localise

the areas for the tool path smoothing. Based on a given tool path the joint motions are

optimised in order to increase the resulting feed rate.

Lavernhe et al. [55] optimises tool trajectories and their follow-up during machining by

orientation smoothing while ensuring the requested accuracy of the machined part. The

optimisation is performed using a surface model for both the tool path and the orientation

with the focus on finding the best orientations so the kinematic performances of the axes

are optimised.

Also by the use of a surface model Grigoriev et al. [36] calculates the tool path based

on surface informations of the workpiece, whose surface is described by Splines. Wang et

al. [89] optimises the cutter contact data within the tolerance and the orientation along

the tool path in order to minimise the tilt angle of the tool. Based on NURBS-surfaces

Qiao et al. [67] derives dual NURBS curves for 5-axis machining. The curves are finally

subdived into Bezier-Splines. Dual curves for tool path planning were also evaluated by

Rufeng [69] by the use of a cubic B-Spline under the condition of smoothness of the tool

path.

An algorithm for the local corner rounding for 5-axis machining is presented by Beudaert et

al. [92]. In order to obtain a continuous path of both the tool tip and the tool orientation a

local rounding of tangency discontinuities between adjacent linear NC-Blocks is performed.

The path of the tool tip and the tool orientation is described by two B-Spline curves, which

are connected with the initial tool path. For the sake of a decrease of the velocity collapses

at the connection of the orientation a parameterisation spline is computed. This leads to a


better feed rate at the tool, but both the kinematic and dynamic behavior of the machine

tool are not taken into account.

Farouki et al. [26] presents an algorithm for the optimisation of the tool orientation of a

5-axis machine tool. In this algorithm the deviation of the tool orientation and the surface

normal is minimised to the constraint of maintaining a constant cutting speed with a ball-

end tool. This leads to an increased productivity for 5-axis machining due to the reduction

of an unnecessary actuation of the machine rotary axes.

Over and above the scientific approaches described before the CNC manufacturers are in-

vestigating their own approaches for the geometry optimisation. A challenging condition

for all the commercial approaches for both geometry and feed rate generation is the re-

quested real time ability. Therefore in contrast to the optimisation approaches illustrated

above usually the geometry is prepared by the use of properly chosen assumptions. Docu-

mentations of algorithms used by the CNC-manufacturers are barely available. Depending

on the manufacturer the smoothing of non tangential transitions is done with circular arcs,

Splines, polynomial functions or filtering.

The CNC controller Indra-Motion MTX 09VRS, which is manufactured by Bosch [11] uses

circular arcs or polynomial functions for the rounding of the block transitions.

Heidenhain [24] has different filtering techniques for the smoothing of a given geometry,

which are parameterised in response to the requested manufacturing result. This leads to

an either rough, smooth or a fast manufacturing process.

Depending on the feed rate of the two segments of the corner, the corner angle and the

time-constant of the exponential acceleration/deceleration Fanuc [31] calculates the feed

rate along the corner.

In a comparable manner Siemens [77] has investigated smoothing algorithms within the

840D CNC controller. The functions G64X lead to a smooth transition of the NC-Blocks

via the insertion of circular arcs, and polynomial functions up to fifth degree.

Due to the fact that most of the NC programs for free-form surfaces consist of short straight

lines and circular arcs, the compression of multiple NC-Blocks is another important task

for the manufacturers of CNC-units. Especially in the case of mould and die applications

with a high demand on the tolerances the quality of the manufactured surfaces relies on a

smooth trajectory with the inclusion of adjacent tool paths within the demanded tolerances

as well as the avoidance of gouging. Like the rounding of the block transitions every CNC

manufacturer has own methods for the compression of multiple NC-Blocks. The user is

supplied with certain parameters but the used method remains hidden in the software.


Surface machining: The main challenge for the geometry optimisation in the field of

surface machining is the calculation of a tool path, which leads to a smooth surface and

a minimum amount of subsequent machining. Especially methods for the optimisation of

the 5-axis milling are further developed in the last years. A widespread overview on 5-axis

flank milling methods is given by Harik et al. [42]

An optimisation of the tool path and orientation within the given tolerance for 5-axis

flank milling is shown by Zhu et al. [99, 98]. Knowing the geometry of the the cutter and

the desired surface the geometry of the tool path and the shape is optimised in order to

increase the accuracy of the machine tool. The kinematic and dynamic behavior of the

machine tool is not taken into account.

By the use of the envelope surface of the tool Lartigue et al. [51] calculate the tool path

deformation in the case of 5-axis flank milling. The tool path and the tool orientation

are described using two curves at particular points of the tool axis. Within the proposed

method the distance between the envelope surface and the desired surface is minimised in

order to increase the machining accuracy.

Wu et al. [91] obtained the tool orientation by two surfaces, which are generated by the

CAD/CAM. A connection-line between these surfaces represents one possible orientation

of the tool path with a certain machining error. A dynamic programming approach leads

to an optimal combination of tool positions with the machining error as objective function.

Lauwers et al. [53] shows the optimisation of the tool inclination angle by matching the tool

profile of a flat end cutter with a temporary spline representing the curvature of the part

surface in the tool contact point. Also based on integrated material removal simulation

Lauwers et al. [54] present a 5-axis milling tool path generation with dynamic step over

calculation.

The automatic generation of an osculating-optimal collision free tool path for five-axis

milling of free form surfaces is shown by Schnider [71].

Pocket machining: The major aim in the case of pocketing is to remove as much

material as possible within a minimum amount of time with respect to the demanded

manufacturing tolerances as well as the strength of the tool.

The optimisation of the tool path for pocket machining for convex pockets without islands

is illustrated by Vosniakos et al. [88] in order to maximise the cutting rate. Based on the

geometry of the requested pocket geometry as well as the radii of the used tools a polygon

for the tool path is evaluated. The optimised geometry is given to the machine tool via

NC-Code. The curvature of the tool path as well as the continuity are not taken into


account. Pateloup et al. [64] show an approach for the optimisation of the tool path of

pocketing by calculation of corner radii according to the kinematic behavior of the machine

tool. Trochoidal tool paths for multi axis milling are optimised by Rauch et al. [68]. The

optimised geometry is given to a Siemens 840D open loop controller via NC code by the

use of the CSPLINE function according to Siemens [77]. Stori et al. [80] present a spiral

in algorithm for 2 1/2D material removal of convex geometries.

Collision avoidance: Especially in the case of 5-axis machining collisions of the work-

piece and the machine tool are critical to handle due to the complexity of the axes move-

ments. Not only for the purpose of collision avoidance but also for the tool path optimi-

sation Castagnetti et al. [17] introduce the domain of admissible orientations (DAO). An

accompanying pyramid restricts the possible orientations of the tool axis for each point of

the tool path for the subsequent smoothing. Also Abele et al. [1], Kanda et al. [49] and

Takeuchi [82] present possible methods and solutions to prevent collisions and collision

damages of machine tool components. The generation of collision-free 5-axis tool paths

by the use of a haptic surface is shown by Balasubramaniam et al. [6]. Lauwers et al.[52]

reveals an optimal and collision free tool posture in five-axis machining through the tight

integration of tool path generation and the simulation of the kinematic of the machine

tool.

2.1.3 Feed rate optimisation for machine tools

Based on the optimised geometry the feed rate optimisation can be seen as subsequent

task for the set point generation. Especially the feed rate optimisation along a curved

tool path is important for machine tools and is regarded below. In order to increase the

productivity the most important goal for the feed rate optimisation is the minimisation

of the machining time. Regarding the quality of the machined parts the mechanical ex-

citation of the machine tool should also be minimised. According to Davison et al.[21] and

Geering [33] the resulting challenge is a minimum time movement without the violation of

predefined boundaries, which is a long known problem in the field of optimal control.

Based on simulations Yazar et al. [93] calculate the cutting force of the 3D-milling process.

Assuming that the process variables are directly influenced by the cutting force, this force

is used as feedback variable to adjust the feed rate. As already mentioned in section 2.1.2

Yu [94] presents basic algorithms for the optimisation of the geometry as well as for the

generation of jerk controlled feed rates.

An indirect method for the feed rate optimisation by the use of a phase plane approach


is shown by Hadorn [38, 39]. Along a one dimensional path parameter the optimisation

algorithm searches for switching points in the phase plane to get the time optimal solution,

which is found when the velocity along the given path is maximal with respect to the given

boundaries. A time optimal solution is obtained by choosing the constraint that leads to

the highest velocity along the tool path. The here arising problem of infeasible regions for

the switching conditions is treated by Hadorn [38], Huang et al. [43] and Shin et al. [76].

A feed scheduling strategy for the purpose of an optimal metal removal rate in 3-axis

machining with respect to the machining accuracy is investigated and validated by Tounsi

et al. [84, 83]. In order to realise a nearly constant cutting force different constraints like

the cutting force, the feed rate boundaries and the contour error are taken into account

during the optimisation process.

Based on geometric constraints Annoni et al. [5] presents a segment wise feed rate limi-

tation with respect to the desired chord error and jerk as well as acceleration constraints.

Based on this feed rate profiles for the different segments are calculated using a windowing

technique.

Methods for both the reduction of mechanical excitation and also the minimisation of

the cycle time along splined tool paths are shown by Altintas et al.[2]. In this approach

the Spline is divided in different segments, whose duration is minimised with respect to

velocity, jerk and acceleration limits. Additionally in every iteration step the integral of

the squared jerk is minimised.

Feed rate interpolation by the intersection of the given constraints is presented by Beudaert

et al. [9]. The aim of the algorithm is to calculate the next reachable point along a given

path within a fixed time-step ∆t knowing all the characteristics on the previous points by

dichotomy.

The time optimal feed rate along a given tool-path with nonlinear constraints is formulated

as an optimal control problem by Steinlin [79]. With the proposed algorithm discontinuities

in the force profile of the trajectory are limited as well as mechanical excitations by the

limitation of the jerk and the jerk-rate.

Stating that a reduction of the jerk limit will not guarantee a decreased mechanical ex-

citation Fauser [27] presented a frequency based approach, which optimises the spectrum

only by a modification of the feed rate profile. Subsequent to the calculation of the max-

imum feed rate for different path segments an acceleration profile with a limited bandwidth

is calculated.


2.2 Feed rate optimisation for redundant machine tools

The use of redundant axes configurations, where a slow and a fast axis arranged in a

congruent manner bares high potential with respect to resulting productivity at a moderate

increase of effort. According to Neugebauer [59] the redundancy of a mechanical system

can be divided in redundant kinematics respectively actuation redundancy and kinematic

redundancy, which are defined and illustrated in figure 2.3.

q1

q2

(a) actuation redundancy

q1

q2q3

(b) kinematic-redundancy

Figure 2.3: Redundant kinematics lead to additional forces on the joints of the system

without an increased number of the degrees of freedom of the end effector (a). Due to the

kinematic-redundancy there is no bijective mapping between the position and the orienta-

tion of the end effector and the different joints. The degrees of freedom of the end effector

are not increased(b)

The motion planning for kinematic redundant systems is an important topic for both

robot applications and machine tools for example for the avoidance of obstacles as well as

singularities. Most approaches for the resolution of the actuation redundancy are published

in the field of robotic science. For example Dasgupta et al. [20] present a trajectory

optimisation of a redundant robot in order to get the shortest path for a point to point

movement by the avoidance of obstacles. For the optimisation of the joint trajectories of

an industrial robot with redundant degrees of freedom a generic approach is presented by

Vosniakos et al. [87]. By the use of a kinematic and dynamic simulation model of the robot

the total work for the motion is minimised. Also Boudreau et al. [13] show an improvement

of the force capabilities by the use of redundancy.

A detailed survey of redundant kinematics for machine tools is given by Steinlin [79] as

well as a quadratic programming approach for the resolution of actuation redundancy for

a 2-axis laser cutting machine tool. By the use of filtering methods using a FIR-filter also

Brecher et al. [16] separate a high from a low dynamic trajectory in the case of a planar

machine tool.

2.3 Error modelling 15

Regarding hybrid redundant kinematics, which consist of both a serial and a parallel

kinematic Schroder [72] and Neugebauer et al. [60] present approaches for the partioning

of motion in order to take advantage of both the large work space of the serial kinematic

and the high dynamics of the parallel-kinematics. Also Harib et al. [41] are splitting the

movement of hybrid redundant kinematic machine tool by a rule-based approach.

Chin et al. [18] illustrates the optimisation of the movement of a redundant kinematic by

the use of a generic approach at the example of parallel kinematic and a redundant cross

table. Generic algorithms (GA) work in a comparable way as the natural selection. An

arbitrarily chosen first generation of a possible solution is evaluated with a fitness function.

In this paper for the motion coordination between the arising redundant degrees of freedom

the fitness function is represented by the summation of the square displacement of each

moving axis of the parallel structure due to the fact that the cross table has a better

tracking precision. As a consequence of that least motion of the parallel structure is

achieved. After that a new generation of solutions is obtained and evaluated by selection,

reproduction, crossover, and mutation. This procedure is repeated until the maximum

generation is reached.

An example of the optimisation of the tool path of a machine tool with redundant degrees of

freedom is shown by Halevi et al. [40]. Exploiting the redundancy the energy consumption

is considerably reduced. This is done by minimising a cost function based on the energy,

which is calculated by a full electro mechanical model of the regarded machine tool.

At the example of a 7-axis fiber placement machine tool Debout et al. [22] demonstrate the

reduction of the manufacturing time while ensuring the quality of the final part by taking

advantage of redundant kinematic degrees of freedom. The smoothing algorithm is based

on a filtering method and incorporates the geometry of the movement of the different axes.

The proposed method minimises the curvature and takes into account the difference of the

dynamic characteristics of each axis in the objective function. The optimised geometry is

described as a polygon, which is later compressed inside the CNC to ensure a continuous

movement.

2.3 Error modelling

For the geometry optimisation a tolerance domain must be defined in a way, that the

resulting error at the TCP, which consists of the systematic deviation due to the smoothing

of the trajectory and the machining errors complies with the requested accuracy.

Experiences show, that the accuracy of machine tools is significantly influenced by different


types of machining errors, which are listed in table 2.1. Due to the fact that this work

is focused on laser cutting machine tools without any process-forces, the process-error

is not incorporated in table 2.1 since it depends strongly on the regarded process and

the specification of the regarded machine tool e.g. number of cutting edges, tool-length,

manufacture material etc....

All these errors lead to a resulting error of a given tool-path. An approach for the iden-

tification of the different types of error is shown by Andolfatto et al. [4]. The tracking

errors are evaluated by comparison of the machine controller inputs and the actual en-

coder values. Because of the usually high stiffness of machine tools at low velocity and low

inertial forces, the dynamic errors at low feed rate are neglected in this approach. Know-

ing the contouring errors and assuming a low feed rate the remaining error contributors

are the quasi-static errors, which are decomposed into the link errors consisting of both

component and location errors of the machine and the thermal drift of the machine.

The contribution of the link errors to the volumetric error along a given tool path is

calculated by a an error model, which is proposed by Zargarbashi et al. [96]. Having a

thermal state of the machine with a negligible thermal drift and a low feed rate all the

resulting errors, which are not explained by the link errors are attributed to the motion

errors and dynamic errors. The effect of the thermal drift on the volumetric errors is

modeled as an offset from the reference thermal state on the measured volumetric errors.

The dynamic geometric errors are defined as additional errors occurring when programmed

feed rates, and so dynamic forces on the machine structure increase.

There are two known ways for the improvement of the accuracy respectively the reduction

of machining errors:

1. error minimisation

2. error compensation

For the minimisation of the inacurracies error avoidance is usually done during the design at

Kind of error Time constant Magnitude µm

component errors ∞ 1− 20location errors ∞ 10− 50thermal errors min - h 1− 100structural dynamic errors 0.2− 1s 5− 20tracking error 0.1s 10− 300

Table 2.1: Different types of machining errors

2.3 Error modelling 17

the manufacturing stage of a machine tool. However, this approach involves a high degree

of investment as machine costs rise exponentially with the level of accuracy according to

to Slocum [78].

Due to the quasi-static behavior of both the geometric and the thermal errors and the

fact, that these error types are the largest sources of inaccuracy, in literatur several com-

pensation approaches can be found.

The prediction of geometric errors and thermal errors is shown by Okafor et al. [62]. For the

example of a 3 axis machine tool and the use of homogeneous coordinate transformation

a mathematic model for the prediction of the resulting volumetric error is developed in

order to compensate it in a final step. A capacitance sensor based multi degree of freedom

(DOF) measurement system for the simultaneous measurement of five error components

is shown by Lee et al. [56]. Here the error estimation is done with a least squares fitting

method in order to represent the geometric errors. The compensation is done with a

recursive compensation approach. The geometric error compensation by a model based

feed forward approach is shown by Gao [30].

Zhu et al. [100] illustrate the error estimation by regarding the machine tool as a multi-

body system and the use of homogeneous transfer matrices. The compensation is done

by NC-Code. Further approaches for the modeling of geometric errors are presented by

Ferreira et al. [28], Treib et al [85, 86] and Yu et al. [95].

For the compensation of thermal errors Ess [25] presents a virtual machine prototype

consisting of a multi-body dynamics and thermo mechanical FE code for the simulation of

thermal errors along a given NC path. The virtual machine prototype uses the NC path

to compute the state of the machine tool and then sends a command back to the CNC to

move the axes in such a way, that no relative error between the TCP and the workpiece

occurs.

Although these compensation strategies and other procedures for the minimisation of the

machining errors increase the accuracy of a machine tool there remains an uncertainty

along the tool path. Getting back to the tolerance domain for the geometry optimisation

usually the remaining uncertainty mentioned above is represented by a global tolerance

value, which is based on the worst case of the remaining uncertainty within the workspace.

Assuming that the geometric errors, which are defined in ISO230-1 [46] are the largest

non-thermal sources of inaccuracy a modelling of this position depend uncertainty should

lead to a position dependent tolerance domain and thus an increased exploitation of the

given manufacturing tolerances. Sellmann et al. [75] presents a tolerance model for the

calculation of the available smoothing tolerance, which is discussed in chapter 6.


2.4 Research gap

Referring to chapter 1 both the machining time and the accuracy of machine tools, which

is directly influenced by the structural excitation are having an impact on the machining

productivity. Assuming a machine tool without any process-forces, e.g. laser-cutting

the mechanical excitation of the machine tool is caused only by the trajectory. Taking

into account, that discontinuous actuator-forces cause structural excitation the machining

productivity can significantly be influenced by trajectories, which lead to continuous forces

and a minimum amount of machining time within certain constraints. These constraints

include point- and axe wise limits for the tool path and the tool orientation along the tool

path as well as axe wise limitations of the acceleration, jerk and the jerk-rate.

Many state of the art methods in the field of geometry optimisation are investigated for

the purpose of surface machining, which is a very challenging task because very small

tolerances and adjacent tool paths have to be taken into account. The appliance of these

methods on laser cutting machine tools leads to a reduced exploitation of the usually

high tolerances for both the tool path and the tool orientation. As a consequence of

that the resulting trajectories have high curvatures, which leads to a higher structural

excitation. Taking into account that the subsequent calculated feed rate is influenced by

the parametric derivatives of the geometry, there is also a need for a global optimisation of

these derivatives with respect to the tolerances for the tool path and the tool orientation.

Regarding 5-axis machining state of the art approaches do not deal with the arising quasi-

redundant degrees of freedom, which are defined in chapter 5 and their exploitation. Espe-

cially the smoothing of the movement of the machine tool axes with respect to the different

inertias of the axes is not taken into account.

Both commercial- and scientific approaches for the geometry optimisation are based on

the assumption of a global conservative value for the smoothing tolerance, which is usually

chosen in order to assure a given accuracy with respect to the known errors of the machine

tool. These errors depend on the position of the tool. The resulting global tolerance

value is evaluated with respect to the worst case behavior of the machine tool inside of

the workspace so there is a demand for an individual tolerance model for the position

dependent calculation of the path tolerance of a machine tool.

The deficiencies described for the optimisation of the geometry and the feed rate for

machine tools with- and without quasi-redundancies as well as the position dependent

calculation of the geometric errors of a machine tool are approached in this thesis.

19

Chapter 3

Fundamentals of curves

In this chapter the mathematical methods, which are used in this thesis are described.

General remarks are given in section 3.1. Section 3.2 presents important properties of

the curves used for the trajectory representation. The representation of orientations is

explained in section 3.3. Common methods for the interpolation and approximation and

their different features are shown in section 3.4. A contemplation on the path dynamics

and the continuity of trajectories is given in section 3.5 and section 3.6.

3.1 General remarks

For the description of the trajectories in this work two coordinate systems have to be

distinguished: A movement of the Tool Center Point (TCP) depending on time t or para-

meter s in the following denoted as rT CP (t) and rT CP (s) describes a relative movement of

the tool and the workpiece in a workpiece fixed coordinate system. The movement of the

different axes of the machine tool in the machine coordinate system (MCS) with respect to

time t or the parameter s is in the following denoted as rMCS(t) and rMCS(s). Both TCP -

and MCS-trajectories are described with parametric curves with respect to an increasing

parameter t respectively s.

The following remarks are based on Aminov [3]. According to this, an d-dimensional

parametric curve is defined as:

Definition 3.1 A parametric curve in Rd is the map r : I → Rd of the closed interval

I := [a, b] to Rd.

Thus every parametric value, in the following denoted as s, inside of the closed interval

I := [a, b] is mapped to a point in Rd. The mapping of the interval [a, b]→ [c, d] allows for

20 3. Fundamentals of curves

a reparameterisation of r(s), which does not affect the shape of the curve in general but

leads to a different mapping of a specific parameter value to a point in space. Moreover,

the derivatives of r(s) are influenced by reparameterisation.

3.2 Important properties of curves

This section briefly describes the relevant characteristics of curves in space, which are

used in this thesis. Important parameters are the parametric derivatives r(i)(s), which are

defined as follows:

Definition 3.2 The ith parametric derivative of r(s) is defined as r(i)(s) = di

dsi r(s). r′(s)is called the parametric velocity, r′′(s) is called parametric acceleration.

The regularity of curves is defined as:

Definition 3.3 r′(s) 6= 0 ∀s ∈ I := [a, b]

Non regular curves are not considered in this thesis, since the feed rate optimisation

will not be feasible in the case of zero parametric velocity. Using the Frenet formulas the

parametric derivatives of a curve in R3 lead to the Frenet frame θ(s) = {et(s), en(s), eb(s)},which is defined as follows:

et(s) = r′(s)|r′(s)| (3.1)

en(s) = (r′(s)× r′′(s))× r′(s)|(r′(s)× r′′(s))× r′(s)| (3.2)

eb(s) = en(s)× et(s) (3.3)

The Frenet frame of a curve is shown in figure 3.1.

et(s) and en(s) are spanning the osculating plane. In this plane lies the curvature circle

as shown in figure 3.1. The rectifying plane and the bi normal plane are perpendicular

to the osculating plane and are also denoted in figure 3.1. While et(s) is tangential to

the curvature circle, en(s) is perpendicular to it and denotes the direction of a connecting

line between the point on the curve and the center of the curvature circle. The reciprocal

value of the radius of the curvature circle is called the curvature κ and describes the rate

of change of the direction of et(s) along the arc length of the curve.

3.3 Orientation representation 21

XY

Z

curvature circle

osculating plane

et(s)

en(s)

eb(s)

Figure 3.1: Frenet frame, osculating plane and curvature circle of a parametric curve.

κ = |r′ × r′′||r′|3 (3.4)

If r′′(s) vanishes, the curvature is zero in this point. If r′′(s) = 0 ∀s ∈ I := [a, b] the

curve is a straight line. The Frenet frame is not defined in this case and can be calculated

according to Hadorn [38] by regarding the neighboring NC-Blocks.

An NC block itself is defined as follows:

Definition 3.4 A NC-Block is a regular curve r(s) according to definition 3.1, which is

defined within the interval 0 ≤ s ≤ 1 and is supposed to be smooth, i.e. all parametric

derivatives are continuous.

3.3 Orientation representation

The methods, which are presented in this work are exemplified on a 5-axis laser cutting

machine tool whose kinematic is sketched in figure 3.2. Quaternions would be the most


general description of the orientation due to the fact that no gimbal lock occurs. Assuming

that the orientation representation in this context is used for the evaluation of the tool

orientation n(sa) as a function of given tool angles ϕB(sa) respectively ϕC(sa) according

to figure 3.2 the gimbal lock is not taken into account, so the tool orientation is sufficiently

described using the rotation scheme (3.5) assuming an initial orientation of the tool n0 =[0 0 1]T and initial rotation angles ϕB(s = 0) = 0 and ϕC(s = 0) = 0.

A first rotation of n0 around the Y -axis of the machine coordinate system by ϕB and a

second rotation around the Z-axis of the machine coordinate system by ϕC leads to the

tool orientation at sa, which is defined as

n(sa) =

cos(ϕC(sa)) sin(ϕC(sa)) 0sin(ϕC(sa)) cos(ϕC(sa)) 0

0 0 1

cos(ϕB(sa)) 0 − sin(ϕB(sa))0 1 0

− sin(ϕB(sa)) 0 cos(ϕB(sa))

001

=

cos(ϕC(sa)) sin(ϕB(sa))sin(ϕB(sa)) sin(ϕC(sa))

cos(ϕB(sa))

(3.5)

As well as the TCP-Position the tool angles ϕB(s) and ϕC(s) depend on the same global

path parameter s, so a position-rotation movement of the machine tool in R3 space is

defined in a similar way.

For the optimisation of the orientation along the tool path in the case of 5-axis machining,

which is shown in the chapters 4 and 5 the movement of the axes of a machine tool rMCS(s),their valuation and thus the transformation TCP →MCS is important.

For the kinematics of the laser machine tool used as example here shown in figure 3.2 the

transformation is defined as:

rMCS(s) = rT CP (s)

−

cos(ϕC(s)) − sin(ϕC(s)) 0sin(ϕC(s)) cos(ϕC(s)) 0

0 0 1

·

− sin(ϕB(s))

0−cos(ϕB(s))

·∆B

+

− sin(ϕC(s))cos(ϕC(s))

0

·∆C

(3.6)

with ϕB(s), ϕC(s), ∆B and ∆C according to figure 3.2.

3.3 Orientation representation 23

XY

Z

rT CP (s)

∆B

∆C

rMCS(s)

ϕB

ϕCn(sa)

Figure 3.2: Example of the kinematics of a laser cutting machine tool.

For kinematics with a larger number of joints standardised procedures for the reverse

kinematic such as Denavit-Hartenberg transformation as shown by Craig [19] are suitable.

Although for a rotational axis the curvature is not defined the parametric derivatives

r(n)(s) are subject of the geometry optimisation and are optimised in chapter 4 in order

to minimise fast changes in orientation movements of the tool.


3.4 Methods for interpolation and approximation

Interpolation and approximation contains methods by which a given function F (s) or

data points (si, f i) can be approximated by a continuous function r(s). According to

Schwarz [73] in the case of an approximation r(s) is determined so that a norm of the

difference between the values of the sought function and the given data points

∥∥∥r(si)− fi

∥∥∥ (3.7)

or the given function and the sought function ‖r(s)− F (s)‖ is minimised. During in-

terpolation the approximating function is constructed so that it coincides with the given

function or the data points at given parameter values:

r(si) = F (si) or r(si) = fi

(3.8)

In literature different approaches for interpolation and approximation can be found. The

Polynomial interpolation, B-Splines and the Non Uniform Rational B-Splines in the fol-

lowing denoted as NURBS are illustrated below because of their practical significance. For

further details it is referred to Schwarz [73], Bartels [7] and Piegl [65]. Important for the

selection of a suitable interpolation method are the requirements, that should be met in

the geometry optimisation. These are:

accuracy: r(si) should represent the fi

within a certain tolerance.

intuitiveness: There should be a relationship between the modification of singular values

of the curve, and the appearance of the curve, in the following denoted as shaping.

continuity: r(s) should be as much continuous as requested.

practicability: A direct application on a commercial NC-unit with respect to the max-

imum polynomial order must be possible.

3.4.1 Polynomial interpolation

Interpolation is used in order to determine new data point within the range of a set of

known data points. A generalisation of the linear interpolation is the polynomial interpo-

lation, which is searching for a polynomial of degree p

r(s) = a0 + a1x+ . . .+ apxp, (3.9)

3.4 Methods for interpolation and approximation 25

which matches the interpolation condition

r(si) = fi, i = 0, 1, . . . , n. (3.10)

According to Schwarz [73] this leads to:

Definition 3.5 To n + 1 arbitrary value pairs (si, f i), i = 0, 1, . . . , n, with distinct real

values si 6= sj∀i 6= j there exists exactly one interpolation r(s) with the property (3.10)

whose degree is at most n.

Insertion of (3.9) in (3.10) provides a linear system of equations, which has only one

solution if si 6= sj for all i 6= j.

Runges phenomenon: If the number of nodes si and the degree of the polynomial

functions of the interpolation increases especially at equidistant nodes the quality of the

interpolation according to definition 3.5 drastically decreases. This becomes clear in the

interpolation function of the Runge-function

F (s) = 11 + s2 , s ∈ [−5; 5] (3.11)

with polynomial function with higher degrees, which is illustrated in figure 3.3.

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.5

0

0.5

1

1.5

2

s

f(s

)

Runge function5th order interpolation polynom10th order interpolation polynomsampling points

Figure 3.3: Interpolation of the Runge-function with 5th- and 10th order polynomial

functions.


If the Tschebyscheff nodes, which are closer at the interval limits, are used instead of

equidistant sampling points, the total error of the interpolation can be reduced as shown

in figure 3.4. For the geometry optimisation, the interpolation with the polynomial func-

tions described above is unsuitable because of its shown susceptibility to mathematical

oscillations. B-Splines also consist of polynomial functions but because of the piecewise

definition shown in the following this type of curves is more suitable for geometry optimi-

sation.

3.4.2 B-Splines

An alternative approach to the polynomial interpolation are the piecewise polynomial

functions, the Splines, which are suitable for the approximation- and interpolation of data

points. The following explanations are based on Schwarz [73] and Piegl [65]. The name

”Spline” is originated from the shipbuilding. Here the Spline is a longitudinal slat, which

is fixed at certain points and behaves like a cubic Spline. The Basis-Spline in the following

denoted as B-Spline of degree p is defined as the linear combination of the control points

ai with the ith B-Spline basis function Ni,p of the degree p:

r(s) =m∑

i=0Ni,p(s)ai a ≤ s ≤ b (3.12)

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.5

0

0.5

1

1.5

2

2.5

s

f(s

)

Runge functionInterpolation with equidistant sampling pointsInterpolation with Tschebyscheff-pointsequidistant sampling pointsTschebyscheff-points

Figure 3.4: Interpolation of the Runge-function with Tschebyscheff-nodes.

3.4 Methods for interpolation and approximation 27

B-Spline basis function: The recursively defined ith B-Spline basis functions of degree

p is obtained for p > 1 as

Ni,p(s) = s− si

si+p − si

·Ni,p−1(s) + si+p+1 − ssi+p+1 − si+1

·Ni+1,p−1(s) (3.13)

with

Ni,0(s) ={

1 if si ≤ s ≤ si+1

0 otherwise(3.14)

Knot vector: si are the knots. At every knot the description of the basis function

changes. All si lead to the knot vector u of the length m + p + 1. For the evaluation of

the knot vector it must be distinguished between closed, opened or clamped curves. In

the following only clamped curves are regarded, which interpolate the first and the last

control point. For these type of curves the knot vector is defined as

u = [ a, . . . , a︸︷︷︸p+1

, . . . , um−p−1, b, . . . , b︸︷︷︸p+1

]. (3.15)

Characteristic is the p + 1-fold occupation of the first and last node and the aperiodic

internal nodes.

Properties of B-Spline curves: The above described definition of the B-Splines leads

to the following properties:

affine invariance: It is no difference if the B-Spline r(s) is first evaluated and then

transformed or whether the points ai are transformed and r(s) is generated on the

basis of these control points.

convex hull: Each segment of a B-Splines of order p is inside of the convex hull of the

segment associated with p control points as shown in figure 3.5

local behavior: Modification of the point ai modifies the B-Spline r(s) only in the inter-

val [si, si+p+1) because of Ni,p(s) = 0 for u /∈ [si, si+p+1).

differentiability: At a knot of the multiplicity k a B-Spline is p− k-times continuously

differentiable. Control points according to knots with maximal multiplicity are in-

terpolated. At this point r(s) is C0 continuous.


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Position X [mm]

PositionY

[mm

]

B-Splineai

convex hull

Figure 3.5: A quintic B-Spline with u = [0 0 0 0 0 0 1 1 1 1 1 1] without internal knots,

which is also known as quintic Bezier-Spline and the enclosing convex hull.

In comparison to the polynomial functions described above B-Splines do not tend to unde-

sirable oscillations. However, B-Splines are non-rational polynomial functions and there-

fore cannot represent arbitrary shapes, e.g. conic sections, circular arcs, ellipses, parabolas

and hyperbolas. For this, the concept of the B-Spline curves is extended to NURBS curves,

which are described in the following section according to Piegl and Tiller [65].

3.4.3 NURBS

NURBS are the general form of the Spline-interpolation and approximation. With NURBS

it is possible to describe conic sections, circular arcs, ellipses, parabolas, hyperbolas and

arbitrary shapes. A NURBS-curve of degree p is defined as

r(s) =∑m

i=0Ni,p(s)wiai∑mi=0Ni,p(s)wi

a ≤ s ≥ b (3.16)

3.5 Path dynamics 29

with the m control points ai, corresponding weights wi and the B-Spline basis functions

Ni,p according to section 3.4.2.

a = 0, b = 1, wi > 0 ∀ i and

Ri,p(s) = Ni,p(s)wi∑mj=0Nj,p(s)wj

(3.17)

(3.16) can be formulated as

r(s) =m∑

i=0Ri,p(s)ai (3.18)

Ri,p are the rational basis functions, which consist of the weights and the B-Spline basis

function. wi are additional weights in order to affect the shape of the curve in the following

manner:

• a large wi (>1) pulls the curve to the point ai.

• wi = 1 ∀i leads to the non-rational B-Spline curve.

• wi = 0 implies, that wi has no influence on the shape of the curve.

Properties of NURBS: In addition to the advantages of B-Splines NURBS offer great

capabilities for the shaping because of their rationality and the weights in order to rep-

resent conics and other shapes. The required continuity can be met with NURBS and

the possibility of the conversion of NURBS into polynomials offers an access to standard

NC-units.

3.5 Path dynamics

In the previous chapter different methods for the description of the path function r(s) were

introduced but do not satisfy the concept of a movement with respect to time t. In order

to obtain the effective velocity along a given curve r(s) a functional dependency between

the path parameter s and the time t, s = s(t) is introduced. Using the chain rule this

leads to

Position x(t) = r(s(t)) (3.19)

Velocitiy x(t) = r′(s)s (3.20)

Acceleration x(t) = r′′(s)s2 + r′(s)s (3.21)

for the definition of the effective velocity respectively acceleration with respect to time t.


3.6 Continuity of trajectories

An important parameter for the set point generation in order to get a physical feasible

trajectory and less mechanical excitation of the machine tool is the continuity of the

calculated trajectory referred to Bartels et al. [7]. In the field of parametric curves the

parametric continuity and the geometric continuity have to be distinguished.

A curve r(s) is said to be Cn continuous if the nth derivative dnrdsn is continuous for s ∈ [a, b].

A descriptive example is the movement of a mass along a precalculated path with respect

to time, which must be at least C1 continuous to have a finite acceleration. In order

to realise a smoother motion like the path planing of a roller-coaster higher orders of

parametric continuity are required.

With the concept of geometric continuity according to Bartels et al. [7] a curve or surface

can be described as Gn continuous with the parameter n as a measurement of the smooth-

ness. Assuming a parametric curve r(s) of the R3 in the closed interval s ∈ [a, b] the first

three orders of geometric continuity are described as follows:

G0: The curve is continuous.

G1: Also the direction but not necessarily the magnitude of the tangent vector is contin-

uous. In other words a curve without any crinkle.

G2: Also the direction but not necessarily the magnitude of the curvature vector is con-

tinuous.

Assuming that discontinuous and also high force respectively force rates lead to a me-

chanical excitation, which is an influencing factor on the productivity of machine tools as

shown in chapter 1 at least G2 respectively C2 continuity is necessary for the set point

generation.

31

Chapter 4

Geometry optimisation for machine

tools

This chapter deals with methods for the geometry optimisation based on the methods for

B-Spline approximation, which are introduced in chapter 3. First of all the global and local

optimisation problem are illustrated in section 4.1 and 4.2. An algorithm for the rounding

of NC-Block transitions and the comprehensive rounding of multiple NC-Blocks within a

given tolerance is presented in section 4.3. Assuming that a maximum exploitation of the

given tolerance leads to a decrease of the local curvature of the trajectory, a local higher

feed rate will be possible. This finally leads to an increase of the machining productivity,

which is exemplified in section 4.4.

4.1 Global optimisation problem

As already mentioned in section 2.1 the global task of the set point generation is to obtain

a time optimal movement within both geometric and dynamic constraints and a minimum

amount of mechanical excitation of the machine tool. A coupled approach, which optimises

the geometry and the feed rate in one calculation step, would be sufficient but in favor of

robustness and due to the requested real time ability of common open-loop controllers the

optimisation task mentioned above is splitted into the two subsequent optimisation steps

geometry optimisation and feed rate optimisation. This approach is in fact heuristic and

does not lead to a time optimal movement in general but is a trade off between robustness

respectively real time ability and the productivity.

The resulting task for the geometry optimisation is to smooth a given tool path within

the given geometric tolerances. Due to the fact that a time optimal trajectory cannot be

32 4. Geometry optimisation for machine tools

preserved only by geometry optimisation, the upcoming question is the objective function

for the geometry optimisation. An important parameter for the smoothness of a curve is

the curvature (3.4) respectively the rate of the curvature. It is shown by Hadorn [38], that

an at leastG2 continuous smoothing function with a minimal curvature leads to a decreased

mechanical excitation of the machine tool. This prevents discontinuous accelerations in

the sensitive direction of the tool path so the mechanical excitation of the machine tool is

finally minimised. According to (3.19)-(3.21) the resulting feed rate and their derivatives

with respect to time, which are calculated in a subsequent task are influenced by the

parametric derivatives r(i)(s). Thus, a smoothing of the curvature respectively the rate

of the curvature, which are calculated by the parametric derivatives of a curve has an

influence on the subsequent calculated feed rate according to Beudaert et al. [10].

Regarding the set up of an optimisation problem for the application case the robustness of

the used optimisation algorithm is important. Assuming a feasible solution the robustness

of an optimisation algorithm is influenced by both the computing time and the convexity of

the optimisation problem. In the field of convex optimisation for the linear programming,

the quadratic programming with both linear and quadratic constraints as well as the

second order cone programming high performance algorithms with real time ability are

available. For the set up of these algorithms an objective function with positive function

values in a comparable magnitude leads to an increased speed of convergence. Therefore

in this work a quadratic programming approach is used. In the case of convexity these

types of optimisation problems can later be reformulated as a second order cone program

according to Boyd [14].

Due to the complexity an objective function, which represents the curvature (3.4) does

not lead to a convex optimisation problem. A convex formulation, which is described in

more detail in section 4.3, minimises the parametric derivatives of r(s) along a given set

of NC blocks.

For simplicity the constraints, which ensure the requested geometric tolerances during

optimisation, are formulated in a linear matter, which is also shown in section 4.3.

The optimisation of the tool path is accomplished depending on the NC code, which usually

consists of one of the two following programming types:

• Geometries, which are described by a sequence of straight lines and circular arcs in

order to realise machining operations on the shop floor.

• Freeform curves, which in the majority of cases are described as polygons. Usually

this type of NC program is created by a CAM tool.

The first type represents sequences of comparatively long NC blocks with a constant curva-

4.1 Global optimisation problem 33

ture so discontinuities are only located at the transition, which is regarded during optimi-

sation. Depending on the requested continuity the resulting trajectory for the smoothing

starts and ends on the original tool path in the immediate vicinity of the transition with

respect to the local direction of the tangent and the local curvature.

Contrary to a sequence of long NC blocks the second type consists of NC blocks with a

very small length, which is usually smaller than 1mm. A rounding of the transitions with

the above described method will be very disadvantageous, because every corner has to

be optimised separately. As a consequence of that the curvature will be continuous but

would oscillate along the tool path so there is a need for the global smoothing of multiple

NC-blocks within a given tolerance. In the field of NC programming this type of rounding

is used to save computational power by pooling of several NC-blocks. For this reason,

this methods are also called compression. While the rounding of NC block transitions is

only focused on the transition between two subsequent NC-blocks the challenging task for

the compression is to find a continuous function to represent a sequence of NC blocks as

smooth as possible within a given tolerance according to figure 4.1.

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2.5

−2

−1.5

−1

−0.5

0

0.5

Position X [mm]

PositionY

[mm

] GeometryCompressionRounding

Figure 4.1: Planar geometry consisting of circular arcs and straight lines. The compres-

sion of multiple NC-blocks is denoted by the red line. The rounding of the transition of

subsequent NC-blocks is denoted by the blue line.


4.2 Local optimisation problem

In the following section the resulting local optimisation problem for the geometry opti-

misation is introduced. Like every optimisation problem, a cost function, constraints and

boundaries have to be formulated.

For a given tool path r(s) consisting of the NC blocks r1(s) to rh(s) with 0 ≤ s ≤ 1an at least G2 continuous curve r(s) with a minimum curvature in every point along the

tool path is sought. In the following it is assumed, that r(s) consists of straight lines and

circular arcs. This leads to the local optimisation problem, which is formulated below.

While in the following all given quantities are denoted with an upper bar, the optimised

quantities are denoted with a tilde.

4.2.1 Objective function

With the parametric derivatives r(i)(s) of the smoothing function the objective function is

defined as

J =i=p∑

i=1

∫ sb

sa

ηi(r(i)(s))2ds. (4.1)

ηi are additional weights, which are defined in section 4.3.5.

4.2.2 Constraints

Since the geometry optimisation is separated from the feed rate optimisation no dynamic

limitations of the different axes of the machine tool are taken into account, and the only

constraint of r(s) is the fulfillment of the given manufacturing tolerances. Additionally,

the process uncertainties need to be taken into account, in order to hold the manufacturing

tolerances.

In consideration of the fact that the methods illustrated in this thesis are exemplified on

5-axis laser cutting the constraints for r(s)T CP and the orientation of the laser beam along

r(s)T CP in order to fulfill the given tolerances can be formulated as described below.

Track tolerance: For the definition of the laser specific tolerances along the tool path

r(s)T CP a local coordinate system ψ(s) is defined. In the following it is assumed, that the

programmed orientation of the laser beam is collinear to the normal vector n(s) to the

4.2 Local optimisation problem 35

sheet metal, which is cut during machining and perpendicular to the cutting direction et(s).The accompanying tripod is aligned to the sheet surface and the cutting line by ψ(s) ={eu(s), ev(s), ew(s)}. It must strictly be distinguished from the local Frenet frame θ(s)(3.1)-(3.3) and is defined as

eu(s) = et(s) (4.2)

ev(s) = n(s)× eu(s) (4.3)

ew(s) = n(s) (4.4)

according to figure 4.2. In the case of a corner the parametric speed r′(s)T CP for the

computation of et(s) and ψ(s) becomes undefined. In this case ψ(s) is calculated by

comparison of the neighboring NC-Blocks.

Regarding the laser cutting process two tolerance values must be taken into account. The

tolerance due to the uncertainty of the process must be considered in the design phase of

the manufactured part. For the geometry optimisation the tolerance due to the focusing of

the laser beam is important since the laser cutting process only works within a small range

for the distance between the TCP and the sheet metal. Therefore the process tolerance

∆proc is introduced, which is defined as the limitation of the deviation ∆w of the TCP in

direction of ew(s).

For the fulfillment of the manufacturing tolerance ∆n and ∆t are defined as the limitation

of the deviations ∆u and ∆v of the smoothing function in direction of ev(s) respectively

eu(s) of the given NC block.

All deviations of the tool path, which are limited by the tolerance values described above

are lying in the cuboidal tolerance volume, which is spanned by eu(s), ev(s) and ew(s) as

shown in figure 4.2. Therefore a deviation between r(s)T CP and r(s)T CP is defined as

r(sa) = r(sa)T CP + ∆ueu(sa) + ∆vev(sa) + ∆wew(sa) (4.5)

according to figure 4.2. As mentioned before ∆proc is given by the process tolerance, which

consists of the following to contributors: An uncertainty of the process, which is regarded

in chapter 6 and a tolerance of the process, which allows for a sufficient laser process and

is considered in the following. ∆n and ∆t are limited by the manufacturing tolerance so a

deviation of the global tool path is admissible if


−∆t ≤ ∆u ≤ ∆t (4.6)

−∆n ≤ ∆v ≤ ∆n (4.7)

−∆proc ≤ ∆w ≤ ∆proc (4.8)

is fulfilled in every point.

XZ

Y

eu(s) = et(s)

ev(s) = n(s) × eu(s)ew(s) = n(s)

∆w

∆v

∆u

r(s)

Figure 4.2: Local trihedron ψ(s), which is based on the parametric derivatives of r(s)T CP

and the tool orientation n(s) as well as the admissible deviations ∆t, ∆proc and ∆n of the

smoothing function.

Orientation tolerance: For the definition of the orientation tolerance a pyramid with

a rectangular base containing all the admissible orientations of the tool axis according to

Castagnetti et al. [17] is defined as shown in figure 4.3.

Lead and tilt angle shown in figure 4.3 are constrained by the process and the manu-

facturing tolerance. From a practical point of view collision avoidance has to be taken into

account but is not regarded in this thesis.

The limiting factor for the lead angle is the process tolerance because orientation deviations

in positive or negative direction of the cutting edge are negligible for the quality of the

4.2 Local optimisation problem 37

XZ

Y

eu(s)

ev(s)

ew(s)

tool axis

∆ϕtilt

∆ϕlead

Figure 4.3: Domain of admissible orientations according to Castagnetti et al. [17].

cutting edge. For the tilt angle on the other hand the maximum skewing of the cutting

edges is the limiting tolerance value so in this case the process tolerance is negligible.

4.2.3 Boundaries

The boundaries deal with the start and the endpoint of the regarded curve. Since the

optimised curve should be embedded into an existing NC program the curve should begin

and end on the original curve in the sense of G0 continuity. Although accomplished by the

increasing parameterisation introduced in chapter 3.1, the beginning of the curve should be

located before the end. As defined in chapter 3.6 also a G2 continuous transition between

r(s) and r(s) is requested and should be mentioned in the boundary conditions, which

have to be formulated.


4.3 Discrete geometry optimisation

In the following a method for the optimisation problem, which is figured out in section 4.1

is presented. With this method both the rounding of the transition zones of adjoining

NC-Blocks and the compression of multiple NC-Blocks is accomplished.

The basic idea of the discrete geometry optimisation (DGO), which is presented in the

following, is to minimise the weighted sum of one or more parametric derivatives of the

sought curve evaluated at given parameter values while ensuring the given tolerances,

which are described in section 4.2. The sought curve for this problem is described by B-

Splines, which are introduced in chapter 3. In favor of robustness the method is formulated

as a quadratic programming problem with linear constraints.

First of all general remarks about quadratic programming are made. Then the formula-

tion of the optimisation problem is presented including the set up of the cost function,

the constraints as well as the boundary conditions. Finally, the influence of individual

parameters on the resulting curve is illustrated and a windowing technique is shown.

4.3.1 Quadratic programming problem

According to [57] a quadratic program (QP) finds a vector x, which minimises

J = 12x

THx+ cTx (4.9)

subject to

Ax ≤ b (4.10)

Aeqx = beq (4.11)

If the matrix H is positive semidefinite, then J is a convex function: In this case the

quadratic program has a global minimiser if there exists some feasible vector x satisfying

the constraints and if J is bounded below on the feasible region. If the matrix H is positive

definite and the problem has a feasible solution, then the global minimiser is unique.

For the solution different commercial algorithms are available. A very efficient method

for the solution of a QP-problem is the interior-point-convex method, which is available

for MATLAB [57] and is used for the computations, which are shown in the remainder of

this chapter. According to [57] quadprog performs the following steps in order to obtain

a solution:

4.3 Discrete geometry optimisation 39

Presolve/Postsolve: In a first step the QP is simplified, e.g. redundant constraints are

removed respectively simplified with respect to Gould [35].

Generate initial Point: According to Mehrotra [58] an initial point x0 fulfilling the

given constraints is generated.

Interior point method: Using a predictor-corrector method, which first predicts a step

from Newton-Raphson formula. Then a corrector step is computed.

Multiple Corrections: In order to increase the step size and to improve subsequent

steps multiple corrections according to Gondzio [34] are computed.

Total Relative Error: An additional function in order to measure the feasibility is eval-

uated. If this function grows too large the problem has no solution.


For the quadratic programming approach the cost function (4.1) has to be reformulated

in a way that it is represented by (4.9). Additionally the resulting QP-problem must be

convex respectively the matrix H in (4.9) must be at least positive semidefinite in order

to obtain a convex optimisation problem.

As already mentioned above, the function r(s) for the rounding of r(s) and its parametric

derivatives are defined as B-Splines of degree p. These are given by a knot vector u for

the evaluation of the B-Spline base functions Ni,p and the control points ai for having an

influence on the shape of the curve. Therefore the vector x in (4.9) contains m control

points ai for each axis, which are optimised. Assuming a knot vector u withm+p+1 entities

and a vector s with n discrete parameter values leads to the ith parametric derivative for

a pth degree B-Spline

r(k)(s) =n∑

i=0N

(k)i,p (s)ai = C(k)(s) · a (4.12)

with C(k)(s) including the base functions N(k)i,p (s). In order to realise a good shape of r(k)(s)

both a parameter vector and an appropriate knot vector must be selected. Assuming that

the parameter lies in the range s ∈ [0, 1] three methods for the parameterisation are

described by Piegl and Tiller [65]. For the DGO, the commonly used parameterisation

proportional to the chord length of rT CP (s) is selected. Other methods are equally spaced

parameters, or the centripetal methods, which are also described by Piegl and Tiller [65].

Let L be the chord length


L =n−1∑

i=0|ri+1 − ri| (4.13)

Then it is

s0 = 0 (4.14)

sn = 1 (4.15)

and

si = si + |ri+1 − ri|L

i = 0, . . . , n− 1. (4.16)

Based on the parameterisation described above a suitable knot vector u must be defined.

For this purpose also by Piegl and Tiller [65, 66] the following technique of averaging for

the computation of u is recommended. It is

u0 = · · · = up = 0 (4.17)

um−p = · · · = um = 1 (4.18)

and

un+p = 1p

n+p−1∑

i=i

si j = 1, . . . , n− p. (4.19)

With this method the knot vector reflects the distribution of s and supports a good shape

of the resulting curve, so the C(i) in (4.12) can be evaluated.

Substituting of (4.12) in (4.1) with a given discretisation s and the nth increment ∆sn

leads to

J =p∑

k=1

∫ sb

sa

ηk(r(k)(s))2ds

=p∑

k=1

n∑

i=1ηk(r(k)(si))2

=p∑

k=1ηk(C(k) · ai)2 (4.20)

Therefore H and c in (4.9) are defined as

H =p∑

k=1ηkC

Tk Ck (4.21)

c = 0 (4.22)


In the general case x consists of the control points ai of the smoothing function r(s).

Knowing that xTHx represents the squared sum of a linear combination of parametric

derivatives it is

xTHx ≥ 0 ∀x ∈ R (4.23)

Therefore H is positive semidefinite and the function J(x) is convex assuming that the

linear constraints (4.10) and (4.11), which are defined below support a feasible solution.

4.3.3 Constraints

The set up of the linear constraints in (4.10) and (4.11) is shown in order to ensure that

r(s) satisfies the given tolerances, which are figured out in section 4.2.2. The control points

ai are defined separately for every axis. However, the deviation between r(s) and r(s) is

defined as illustrated in figure 4.2 and 4.3. In the sense of a maximum utilisation, the given

tolerances must be shared by all the axes, which are involved in a given trajectory as well

as the process. Therefore the tolerance-sharing for both the tool path and the orientation

of the tool is presented in the following.

Tolerance sharing for the tool path: During computation of the DGO the original

geometry is discretised by n points. In order to get a feasible solution, the tolerance for

the TCP, which is defined in section 4.2.2 must be satisfied at every point. Therefore the

tolerance sharing for the tool path is accomplished by evaluation of ψ(s) at every regarded

point sa along the geometry r(s). To ensure that the movement of the nth value of r(s)arising from (4.5) is limited to a cuboidal volume spanned by the local accompanying

trihedron ψn(s) with the edge lengths 2∆u, 2∆v and 2∆w, during optimisation (4.5)-(4.8)

must be represented by the linear constraints (4.10)-(4.11).

For the representation of (4.5) the vector x in (4.9)-(4.11) is defined as

x = {Xm, Ym, Zm,∆un,∆vn,∆wn} (4.24)

with the dimension [3m+ 3n, 1]. For the representation of (4.5) it is


Aeq

=

−et,X,1 −en,X,1 −eq,X,1

C(0) 0 0 . . . . . . . . .

−et,X,n −en,X,n −eq,X,n

−et,Y,1 −en,Y,1 −eq,Y,1

0 C(0) 0 . . . . . . . . .

−et,Y,n −en,Y,n −eq,Y,n

−et,Z,1 −en,Z,1 −eq,Z,1

0 0 C(0) . . . . . . . . .

−et,Z,n −en,Z,n −eq,Z,n

(4.25)

and

beq =

rX1...

rXn

rY 1...

rY n

rZ1...

rZn

(4.26)

so the nth value of r(s) is only able to move along the local accompanying trihedron ψ(sa).

C(0) in (4.25) are the base functions of the requested B-Spline evaluated at the n assumed

parameter values si. The other contributors of (4.25) are diagonal matrices of the X, Y

and Z components of the normed components of ψ(s).

(4.6)-(4.8) are represented by the linear inequality constraints (4.10) with

A =

0 . . . 0 I 0 0...

. . .... 0 I 0

0 . . . 0 0 0 I

0 . . . 0 −I 0 0...

. . .... 0 −I 0

0 . . . 0 0 0 −I

(4.27)


and

b =

∆t

∆n

∆proc

−∆t

−∆n

−∆proc

. (4.28)

Knowing the matrix H for the objective function, which was defined in section 4.3.2 and

the constraints defined above the quadratic programming problem (4.9)-(4.11) for the tool

path is sufficiently defined to obtain a feasible solution within the constraints. This is

shown at the example of a simple planar geometry consisting of two linear NC-blocks,

which are connected G1-continuously by an arc segment according to figure 4.4. Using the

constraints, which are defined above the geometry is rounded within a maximum deviation

in normal direction of the tool path of 50µm using the quadratic programming approach

(4.9)-(4.11). From figure 4.5 it is obvious that the tolerance in normal direction of the

tool path is satisfied at every point of the geometry. Due to the fact that the transitions

between the NC blocks are tangent, the tolerance is not exploited like in the case of a

sharp corner.

Tolerance sharing for rotational axes: In the case of 5-axis laser cutting the given

tolerance values for the lead and tilt angle, which are introduced in 4.2.2 strongly differ.

Depending on the process and excluding the avoidance of obstacles, lead angles up to 20◦are admissible during manufacturing. On the other hand the tilt angle is limited by the

manufacturing tolerance. Usual values for the tolerance of the tilt angle are up to 5◦.

Taking into account, that the optimisation is carried out at the TCP for every machine

axis, the orientation tolerance described in section 4.2.2 must be shared by the rotational

axes, which are involved in the trajectory. Because of the nonlinear transformation between

rT CP (s) and rMCS(s) according to (3.6) and the circumstance, that the linear inequality

constraints (4.10) must contain the limitation of both the lead and the tilt angle along a

given trajectory, a method for the formulation of the tolerance sharing for rotational axes

is required. For this purpose the domain of admissible orientations (DAO), which was

investigated by Castagnetti et al. [17] is used.

The procedure for the calculation of the DAO in order to derive linear constraints for the

optimisation of the movement of the rotational axes is as follows according to figure 4.6.

For further details it is referred to Fischer [29] and Castagnetti et al. [17]:

Creation: For every point along the trajectory the DAO is defined in the local part coor-


0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

60

70

80

Position X [mm]

PositionY

[mm

]

Original geometry r(s)Geometrical error of r(s) · 200

Figure 4.4: 2D-Geometry, which consists of 2 straight lines, which are connected G1

continuous by a circular arc (black). In favor of the visibility the deviation of the smooth-

ing function in normal direction of the tool path is scaled two hundred times (red). The

wiggling of r(s) in the immediate vicinity of the transitions of the different NC blocks

does not necessarily lead to a time optimal movement but a smooth trajectory without any

discontinuities.

dinate system ψ(s). In the regarded case of 3D laser cutting, the DAO is described

as a pyramid with a rectangular base containing all the possible orientations of the

tool axis for each point on the tool path.

Transformation: Taking into account the kinematics of the machine tool for every orien-

tation of the tool within the DAO a corresponding tuple of angles for the rotational

axes B and C is calculated using the transformation (3.6). For each point on the

tool path this leads to an arbitrarily area in the plane spanned by B and C.

Linearisation: Choosing a rectangle inside of the arbitrarily area in the B/C plane de-

scribed above, a domain of admissible angles is calculated. The upper and lower

limitation of B respectively C is given by the edges of the chosen rectangular. Based


0 20 40 60 80 100 120 140−60

−40

−20

0

20

40

60

80

Path [mm]

GeometricalError

[µm

]

Geometrical Error [µm]maximum permissible deviation [µm]

Figure 4.5: Deviation in normal direction of the tool path of figure 4.4 along the arc

length.

on this linear constraints for the quadratic programming problem are derived as

described by Fischer [29].

4.3.4 Boundaries

As mentioned in section 4.2 an at least G2 continuous transition between r(s) and r(s) is

requested and should be mentioned in the boundary conditions, which are formulated in the

following. First of all the λ-parameterisation, which was investigated by Hadorn [38] for the

quintic Bezier-Spline is generalised for a B-Spline of degree p. Since the λ-parameterisation

is originated in order to ensure a G2 continuous movement of linear axes an expansion for

rotational axes is presented.

λ-parameterisation for the tool path: A B-Spline curve shall be computed in order

to connect the two given points ra and rb. The local Frenet frames at the begin and the

end of the curve are given as θa respectively θb. Boundary conditions for the curvature

are given by κa at s = 0 and κb at s = 1. In the case of linear axes and the fact that

r(s) is connected with common NC-blocks like straight lines respectively circular arcs the

following demands on the parametric derivatives at the begin


Creation

Transformation

Linearisation

Local part coordinate system (TCP)

Maschine coordinate system (MCS)

Maschine coordinate system (MCS)

B

C

B

C

XZ

Y

eu(s)

ev(s)ew(s)

tool axis

∆ϕtilt

∆ϕlead

Figure 4.6: Workflow for the point wise evaluation of the DAO in order to obtain linear

constraints for the subsequent optimisation.

ra = r(s=0) (4.29)

λaet,a = r′(s=0) (4.30)

κa =|r′(s=0) × r′′(s=0)||r′(s=0)|3

(4.31)

0 = r′(s=0) · r′′(s=0) (4.32)


and the end of the curve

rb = r(s=1) (4.33)

λbet,e = r′(s=1) (4.34)

κb =|r′(s=1) × r′′(s=1)||r′(s=1)|3

(4.35)

0 = r′(s=1) · r′′(s=1) (4.36)

occur, in order to ensure G2 continuity between the smoothing function and the planned

path.

Since only clamped curves defined by the knot vector (3.15) are used in this work r(s)interpolates the first and the last control point so G0 respectively C0 continuity is ensured

by a0 = ra and an = rb.

According to Piegl and Tiller [66] the first two parametric derivatives of a B-Spline of

degree p for s = 0 respectively s = 1 are defined as

r′(s = 0) = p

up+1(a1 − a0) (4.37)

r′(s = 1) = p

1− um−p−1(an − an−1) (4.38)

r′′(s = 0) = p(p− 1)up+1

(a0up+1

− (up+1 + up+2)a1

up+1up+2+ a2up+2

)(4.39)

r′′(s = 1) = p(p− 1)1− um−p−1(

an

1− um−p−1− (2− um−p−1 − um−p−2)an−1

(1− um−p−1)(1− um−p−2) + an−21− um−p−2

)(4.40)

Just as already defined by Hadorn [38] the 6 λ-parameters λ0 . . . λ5 are introduced in

order to ensure a movement of the control points ai at the begin and the end of r(s) on

the osculating plane spanned by the local Frenet frames θa and θb, which is evaluated at

the transition between r(s) and r(s). According to figure 4.7 the control points at the

begin and the end of r(s) are defined as


a1 = a0 + λ0et,a (4.41)

a2 = a0 + λ1et,a + λ2en,a (4.42)

an−1 = an − λ5et,e (4.43)

an−2 = an − λ4et,e + λ3en,e (4.44)

XZ

Y

a0

a1

a2

λ0

λ1

λ2

et,aen,a

eb,a

ra

Figure 4.7: λ-parameterisation for the begin of a B-Spline. As defined in (4.41)-(4.44)

by modification of λ0 . . . λ2 parameters the control points a1 and a2 are moved on the

osculating plane spanned by the local Frenet frame θa.

(4.41)-(4.44) in (4.37)-(4.40) leads to

λ1 = λ0up+1 + up+2

up+1(4.45)

λ2 =κ0up+2λ

20p

up+1(p− 1) (4.46)

λ4 = λ5um−p−1 + um−p−2 − 2

um−p−1 − 1 (4.47)

λ3 =κ1(um−p−2 − 1)λ2

5p

(um−p−1 − 1)(p− 1) (4.48)

for the fulfillment of (4.29) -(4.36). Obviously λ0 respectively λ5 are free parameters for

the shaping of the curve while ensuring the boundaries described above.


λ-parameterisation for the tool orientation: In order to ensure G2 continuity for

the optimised trajectory rT CP (s) respectively rMCS(s) also the continuity of the rotational

axes at the transition between r(s) and r(s) is required.

As already mentioned above rT CP (s) consists of straight lines and circular arcs. In common

open loop controllers for these type of NC-blocks a trajectory for a rotational axis C is a

linear interpolation between the start point Ca and the endpoint Cb along a tool path of

the arc length L.

Therefore in the followingd2C

ds2 = 0 ∀s ∈ I := [a, b] (4.49)

is assumed. Based on this, the λ-parameterised control points of a rotational axis C at

the begin and the end of r(s) are defined as

C1 = C0 + λ0et,c,a (4.50)

C2 = C0 + λ1et,c,a + λ2en,c,a (4.51)

Cn−1 = Cn − λ5et,c,e (4.52)

Cn−2 = Cn − λ4et,c,e + λ3en,c,e (4.53)

with C0 = Ca respectively Cn = Cb in order to ensure G0- respectively C0 continuity and

the λi, which are already defined in (4.41)-(4.44). Due to (4.49) λ2 and λ3 are zero. The

remaining scalar value et,c,a and et,c,e are calculated as follows:

A B-Spline of degree p for a rotational axis C shall be computed in order to ensure

dC

dL (L=0)= ∆C

∆l a(4.54)

with the gradient of the previous NC-block ∆C∆L a

and the first derivative of C(L), which is

defined as

dC

dL= dC

ds

ds

dL= dC

ds

1dLds

= C ′

L′(4.55)

Comparable to (4.37) for a B-Spline of degree p C ′ for s = 0 is defined as

dC

ds(s = 0) = p

up+1(C1 − C0). (4.56)

(4.50) in (4.56) leads todC

ds= pλ0et,c,a

up+1(4.57)


A function L(s) for the description of the arc length L can also be computed as B-Spline

of degree p. Their parametric derivative dLds

for s = 0 is defined as

dL

ds= p

up+1(L1 − L0). (4.58)

With L0 = 0 it isdL

ds= p

up+1L1. (4.59)

The arc length L of a spatial curve is defined as

L(s) =∫ s

0|r′(s)|ds (4.60)

with the parametric derivative

L′(s) = |r′(s)|. (4.61)

For s = 0 (4.61) yields to

L′(s = 0) = pλ0

up+1(4.62)

(4.62) in (4.59) leads to

L1 = λ0. (4.63)

(4.59) with (4.63) and (4.56) in (4.54) leads to

∆C∆L a

= C ′

L′ s=0= et,c,a. (4.64)

The same considerations for s = 1 yield

∆C∆L b

= C ′

L′ s=1= et,c,e. (4.65)

4.3.5 Parameterisation

Although the local optimisation problem is formulated with respect to the given constraints

and boundaries, a couple of parameters still remains undefined. These are the number of

control points, which are used for the smoothing spline, the degree of the B-Spline, which

is used during optimisation as well as the additional weights ηK in (4.1). Their influence

on the optimisation result, especially on the shape of the curve is shown in the following at

the example of the rounding of a planar rectangular corner within a permissible deviation

of the tool path of 50µm.


Influence of the number of control points: Taking into account the local behavior

of the B-Spline, which is described in section 3.4.2, the shape of the resulting curve is

influenced by the number of control points, which are used for the smoothing Spline. In

figure 4.8 different results for the smoothing of the planar geometry, which is described

above are shown in response to the number of control points. The smoothing of the corner

leads to local maxima of the higher parametric derivatives of r(s) in the vicinity of the

corner. It is apparent that a decreased number of control points leads to a more global

behavior of the smoothing spline.

0 10 20 30 40 50 60 70 80 90 100

−20

0

20

40

Path [mm]

GeometricalError

[µm

]

r(s) with 1000 pointsr(s) with 800 pointsr(s) with 600 pointsr(s) with 400 points

Figure 4.8: Deviation in normal direction of the tool path in response to the number of

control points.

Influence of the additional weights ηk: For the weighting of the different parametric

derivatives of r(s) the additional weights ηk in (4.1) are used. As already mentioned in

section 4.1 the global task for the geometry optimisation is the smoothing of a given geo-

metry, which does not lead to a time optimal movement in general but leads to a decreased

mechanical excitation of the machine tool due to the elimination of discontinuities and the

minimisation of the curvature. For the example of a quintic B-Spline their influence on

the resulting geometrical error due to the smoothing of the planar geometry, which is

described above, is shown in figure 4.9. For simplicity, only the following tuples are used:

η1 = {1, 0, 0, 0}: Weighting of the parametric speed

η2 = {0, 1, 0, 0}: Weighting of the parametric acceleration


η3 = {0, 0, 1, 0}: Weighting of the parametric jerk

η4 = {0, 0, 0, 1}: Weighting of the parametric jerk-rate

0 10 20 30 40 500

10

20

30

40

50

Position X [mm]

PositionY

[mm

]

Original geometryDeviation for η1 · 50Deviation for η2 · 50Deviation for η3 · 50Deviation for η4 · 50

Figure 4.9: Deviation in normal direction of the tool path in response to the tuple of the

used weighting factors ηi.

A minimisation of the parametric jerk and the jerk-rate due to η3 respectively η4 leads to

unintended wiggles on the shape according to figure 4.9. This is caused by the different

order of magnitude of the different parametric derivatives, so small changes of function

values of the higher derivatives lead to high changes of the function values of the lower

derivatives. Thus only a minimisation of the parametric speed and acceleration will provide

a sufficient shape of the resulting curve r(s).

According to Beudaert et al. [10] a minimisation of the second parametric derivative,

which is proportional to the curvature (3.4), leads to a smooth trajectory. Additionally a

parametric study, which is shown in appendix A leads to η = {0, 1, 0, 0} and is used in the

remainder of this chapter.

Influence of the spline degree: The degree of the B-Spline, which is used for the

smoothing of a geometry within the given tolerance has also an influence on the result.

The deviation in normal direction of the tool path along the arc length of the test geometry


is illustrated in figure 4.10. Obviously the maximum deviation in the corner at 50mm is

only reached for a short distance. This is caused by the used weighting factor figured out

above. This leads to a trajectory, which uses both the convex and concave side of the given

track within the constraints. Experiments show that the best shape is provided using a

cubic B-Spline.

0 10 20 30 40 50 60 70 80 90 100

−20

0

20

40

Path [mm]

GeometricalError

[µm]

Quintic B-Spline r(s)Quartic B-Spline r(s)Cubic B-Spline r(s)

Figure 4.10: Deviation in normal direction of the tool path in response to the degree of

the used B-Spline.

4.3.6 Windowing

In order to save computation power and because of the requested ability of rounding large

data sets, windowing techniques are necessary. In this context windowing means, that the

smoothing of r(s) is split in multiple computation parts, in the following named windows,

which are subsequently optimised and linked. The challenging task for the windowing is the

choice of a strategy in order to minimise the deviation between a smoothing function r(s)with and without windowing. Therefore a strategy for the set up of the different windows

is shown. Additionally at least C2 continuity is required, which must be taken into account

during the set up of the constraints of the different windows. Since the optimised trajectory

r(s) is described by clamped B-Splines with internal knots of multiplicity 1, methods for

the splitting and joining of different B-Splines are shown below based on Piegl and Tiller

[65].


Windowing strategy: In the case of an optimisation of huge data sets a common strat-

egy for the windowing is a forward computation. It consists of a computation window,

which is moved along the data set. Upcoming questions are the window length, the tran-

sition between subsequent computation windows as well as the overlapping of subsequent

windows. The window length is limited by the available computing power. In order to ob-

tain Cn continuity for the final trajectory, the starting condition of a computation window

ri(s) is defined by the transition condition located within the previous window ri−1(s).This leads to additional linear boundary conditions

r(k)i (si = 0) = r

(k)i−1(si−1) ∀si−1 ∈ I := [a, b] (4.66)

for the kth derivative at the starting point of ri(s). The end condition is left free.

If the transition between subsequent computations windows is located at the end of ri−1(s),the subsequent part of the trajectory r(s) is not taken into account during the optimisation

of ri−1(s). In the case of an uncertainty in the immediate vicinity of ri−1(s = 1) this leads

to bad starting conditions for ri(s). As a consequence of that subsequent computation

windows must overlap in a way, that the transition is located in a sufficient distance to

the end of ri−1(s). For simplicity and because of the local behavior of the used B-Splines,

a fixed overlapping of 40% is used in the following.

Methods for the splitting and joining of B-Splines: Two basic methods for the

windowing using B-Splines are the knot insertion and the knot removal, which are shortly

described in the following based on Piegl and Tiller [65].

Knot insertion: For the subdividing of a given curve or surface at a given parameter sa

the knot insertion is a common way in the field of NURBS respectively B-Splines.

For the windowing it is very useful, because it is only a change of the vector basis.

The shape of the curve and the derivatives are not affected. For the curve splitting of

a pth degree B-Spline at a parameter sa and in order to obtain a defined transition

point, which is interpolated by r(s), a knot of multiplicity k = p is inserted as

described by Piegl and Tiller [65]. This leads to new basis functions, which are

p − k times continuously differentiable at the new knot sa. In order to ensure the

continuity of the resulting curve, the derivatives of the resulting curve at sa are the

boundary conditions for the subsequent computation window. At the very end of the

optimisation multiple B-Spline of continuity Cp−1 with outer knots of multiplicity

k = p are lined up.

Knot removal: In order to join the B-Splines to one curve which is p times continuously

differentiable at each parameter s a knot removal is carried out. The knot removal is

4.4 Application examples 55

the reverse process of the knot insertion and is used for the removal of interior knots

of multiplicity k which result from the windowing as described above. Assuming

that the B-Splines are connected Cp−1 continuous it is not necessary to check if the

knot is removable, so given algorithms for the knot removal according to Piegl and

Tiller [65] can be simplified and used for this purpose.

4.4 Application examples

In the following section two examples for geometry optimisation are presented and dis-

cussed. First of all a virtual CNC for the evaluation of the feed rate and a machine model

for the simulation of the contouring error are introduced. The first example shows the

influence of the DGO on both the feed rate and the resulting contouring error for a planar

geometry. The second example illustrates the influence of the DGO on the feed rate in the

case of a 3D geometry including the smoothing of both the tool path and the orientation

of the tool along the tool path.

4.4.1 Virtual CNC

Since this chapter only deals with geometry optimisation the feed rate for the illustration

of the influence of the DGO is calculated from the virtual CNC, where the tool path is

introduced and the virtual CNC computes maximum feed rate. The virtual CNC is an

image of Siemens 840D open loop controller of a 5-axis machine tool and is used for feed

rate simulations. The feed rate constraints for the simulation are listed in table 4.1.

In order to obtain a time discrete profile of the feed rate the optimised geometry r(s), which

is available as B-Spline, is converted into NC-Code and given to the open-loop controller.

The execution of the resulting NC-Code leads to the required time discrete profile of the

feed rate, which is given to a machine model for the simulation of the contouring error,

Max. Velocity Max. Acceleration

X-axis 100 m/min 0.9 gY-axis 100 m/min 1gZ-axis 100 m/min 1gB-axis 3π 1/s 200 1/s2

C-axis 3π 1/s 100 1/s2

Table 4.1: Maximum Velocity, maximum acceleration and maximum jerk of the simulated

machine tool


which is described in the following.

4.4.2 Machine model

The machine model, which is explained in the following, is a rigid body model including

a model for the control system according to Nguyen [61].

Regarding a machine tool in the low and medium frequency range shifts primarily take

place in the coupling locations between the mechanical machine components like linear

guide ways or rotary bearings. Therefore the deflection behavior in the low and medium

frequency range is actually given by the deflection of these coupling elements. This allows

for an approach with multi body dynamics, which finally leads to results comparable with

those obtained by Finite Element Methods.

The model structure is coupled with a control system so the machine tool can be simulated

with respect to their structural behavior as well as their control parameters already in an

early phase of the machine tool design. In this context the model, which is explained

above and sketched in figure 4.11, is used for the simulation of the system response of an

existing machine tool.

4.4.3 2D laser cutting example

For the illustration of the influence of the DGO on machine oscillations and the resulting

feed rate, the planar geometry shown in figure 4.12 is rounded with a global tolerance value

in normal direction of the tool path of 50µm. The exploitation of the given tolerance in

normal direction of the tool path is shown in figure 4.13. Due to the selected parame-

terisation the given tolerance is not completely exploited. The optimised trajectory r(s)is given to the virtual CNC in order to obtain time discrete feed rate profiles. These are

given to the described machine model.

Regarding the resulting feed rate profile for the reference geometry, which is shown in

figure 4.14 velocity collapses at the boundaries of the different NC-block occur. Since

the reference geometry is not G2 continuous, the feed rate is reduced at the boundaries

between subsequent NC-blocks. In contrast, the geometry resulting from the DGO is G2

continuous, which provides a better feed rate since the velocity collapse at the boundaries

are significantly reduced. This leads to a time saving of 20%.

In order to evaluate the mechanical excitation of the machine tool the machine model,

which is described in section 4.4.2 is used. The resulting deviation at the TCP consists

of systematic errors caused by the smoothing of r(s), contouring errors, geometric errors


−10

12

34

−10

12

−0.50

0.51

1.52

Position X [m]Position Y [m]

PositionZ

[m]

Figure 4.11: Machine model of the 5-axis laser cutting machine tool.

−20 −10 0 10 20 30 40 50 60 70 80 90 100 110 120−50

−40

−30

−20

−10

0

10

20

Position X [mm]

PositionY

[mm

]

Figure 4.12: 2D geometry, which is optimised with the DGO using the parameters, which

are identified in section 4.3.5.


0 50 100 150 200 250 300 350 400 450−60

−40

−20

0

20

40

60

80

Path [mm]

GeometricalError

[µm]

Geometrical Error [µm]maximum permissible deviation [µm]

Figure 4.13: Programmed deviation in lateral direction of the tool path.

0 0.5 1 1.5 2 2.5 3 3.50

100

200

300∆t = 20%

Time [s]

Feedrate

[mm

/s]

Original geometry r(t)Optimised geometry r(t)

Figure 4.14: Feed rate profiles, generated by the virtual CNC for the original geometry

(red) and the optimised geometry (green). Due to the decreased velocity collapses of the

optimised geometry a time saving of 20% is achieved.

of the machine tool as well as thermal errors, process errors and dynamic errors. The

dynamic errors are separated by evaluation of the spatial difference between the internal

measurement system and the TCP. As can be seen in figure 4.15 the maximum spatial

deviation between the TCP and the internal measurement system and therefore the me-


chanical excitation can be reduced through optimisation.

0 50 100 150 200 250 300 350 400 4500

5

10

15

20

25

30

35

40

47% reduction of the

maximum geometrical

error

Path [mm]

GeometricalError

[µm

]


Figure 4.15: Spatial deviation between the TCP and the internal measurement system.

Due to the optimisation with the DGO the maximum geometrical error can significantly be

reduced

4.4.4 3D laser cutting example

In the following the influence of the DGO on the resulting feed rate is presented for the ex-

ample of a typical geometry in the field of 3D sheet metal cutting shown in figure 4.16. The

geometry especially the trimming cut of the edge of the sheet metal shown in figure 4.17

is a very illustrative example because of the fast changes in the orientation of the tool

along the tool path. Depending on the kinematic of the machine tool these orientation

movements lead to large compensation movements, which are also shown in figure 4.17.

A rounding of both the tool path and the orientation within the given tolerance should

provide a smooth movement of the machine tool and thus a decrease of the dynamic error


Figure 4.16: CAD visualisation of the regarded sheet metal including the the desired

orientation of the laser beam (red) along the trimming cut. Because of the high curvature

of the sheet metal fast changes of the orientation along the tool path in order to ensure a

stable laser process are required.

−2000

200400

600800

1,000

−400−200

0200

−200

0

200

Position X [mm]Position Y [mm]

PositionZ

[mm

]

rT CP (s)rMCS(s)

Figure 4.17: rT CP (s) (red) and rMCS(s) (blue) of the trimming cut of the sheet metal

shown in figure 4.16. Obviously fast changes of the orientation along the tool path lead to

large compensation movements.


as well as a better feed rate. The smoothing of the trimming cut of the edge of the sheet

metal, which is contemplated in the following, is carried out using global tolerance values

for the deviation of the tool path and the tool orientation of 50µm respectively 5◦ for

both the lead and the tilt angle. Because of the large data set for the description of the

geometry the windowing technique described in section 4.3.6 is used. Due to the fact that

the open loop controller of the virtual CNC is only able to handle a limited number of NC

blocks a description of the optimised geometry with a minimum amount of control points

ai is required. This is obtained by an iterative reduction of the number of control points

while ensuring a valid solution of (4.1) within the constraints. The number of discrete

points of r(s) remains unchanged. As a consequence of that the accuracy of the obtained

solution is kept, while the number of control points is reduced.

The optimisation in this example is subdivided in two computation parts. In a first step

the smoothing of the rotational axes within the constraints is carried out. For the sake

of robustness no tolerance-sharing for the rotational axes is used. Therefore the different

axes are separately optimised. The results for the B respectively the C axis are shown in

figure 4.18 and 4.19.

0 500 1,000 1,500 2,000 2,500 3,000 3,500−100

−50

0

50

Path [mm]

Angle

[◦ ]

rB(s)rB(s)

Figure 4.18: Trajectories rB(s) and rB(s) of the geometry, which is shown in figure 4.16

respectively figure 4.17.

Obviously the smoothing of a large change of the tool orientation along a short arc length at

the TCP leads to unintended wiggles on the smoothing spline rB,C(s). These are caused by

local extrema of higher parametric derivatives r(k)B,C(s), which lead to the described wiggles

as well as compensation movements of the linear axes.

Knowing the orientation of the tool, ψ(s) is evaluated for the tolerance sharing, so different


0 500 1,000 1,500 2,000 2,500 3,000 3,500−100

0

100

200

Path [mm]

Angle

[◦ ]rC(s)rC(s)

Figure 4.19: Trajectories rC(s) and rC(s) of the geometry, which is shown in figure 4.16

respectively figure 4.17.

tolerance values for ∆n and ∆proc can be distinguished during optimisation. The regarded

5-axis laser cutting machine tool is equipped with an additional redundant axis in the

direction of the laser beam, which is used for the adjustment of the gas nozzle, which is

used for the focusing of the laser. Since this axis is controlled with a closed loop controller,

there is no option of user supplied set points for this axis. Therefore the tolerance value

∆proc in direction of the laser beam can be enlarged up to the stroke of the redundant axis

of ±5mm.

The resulting feed rate along the trimming cut compared to the feed rate along the original

geometry in response to the time is shown in figure 4.20. For this example the time saving

only by smoothing of the TCP-path and the orientation of the tool is about 20%. Due to

the fact, that the machine model, which is described in section 4.4.2 is only defined for

small angles for the rotational axes, the mechanical excitation of the machine tool is not

regarded for this application example.

The DGO does not consider the kinematic behavior of the machine tool, so the algorithm

works for every machine tool. If rotational axes are involved in the trajectory, a round-

ing of all the axes does in fact provide a continuous movement of all the axes, but the

resulting MCS-trajectory is inherently not optimal in the sense of jerk- or jerk-rate mini-

malism. Therefore the exploitation of quasi-redundant degrees of freedom is provided and

is discussed in chapter 5.


0 2 4 6 8 10 12 14 16 18 20 220

100

200

300∆t = 20%

Time [s]

Feedrate

[mm

/s]


Figure 4.20: Exemplification of the time saving due to the smoothing of both the tool path

and the tool orientation. Because of the generous orientation tolerance a smooth movement

of the rotational axes and therefore smooth compensation movements and finally a good

feed rate is achieved.

64 5. Feed rate optimisation using quasi-redundant degrees of freedom

Chapter 5

Feed rate optimisation using

quasi-redundant degrees of freedom

This chapter presents two approaches for the exploitation of quasi-redundancy. First of

all the definition of this phenomenon is given in section 5.1. The conception of the opti-

misation for the exploitation of quasi-redundancy is presented in section 5.2. The general

formulation of the arising optimisation problem including the regarded constraints and

boundaries is figured out in section 5.3. Both a nonlinear and a quadratic programing

approach for the exploitation of quasi-redundancy are presented in section 5.4 respec-

tively 5.5. In order to show the increased productivity during machining the effect of the

proposed method is exemplified in section 5.6 and discussed in section 5.7.

5.1 Definition of quasi-redundancy

Redundancy is originated from robotics. Here the treatment of redundant kinematics and

kinematic redundancy are important tasks in the field of robotics control. A redundant

kinematic consists of a main and a second subsystem, which are arranged in a serial

manner, so one position in the workspace can be reached from various positions of the

main and the subsystem. Therefore the set point generation for a redundant kinematic

system has to separate the tasks of the two axis subsystems. Regarding a kinematic

redundant robot there is no bijective mapping between the position of the end effector

and the position of the different joints, which is an important task in the field of inverse

kinematics.

Regarding a 5-axis machine tool consisting of rotational and translational axes the trans-

formation between the TCP- and the MCS movement is unique, but within the given

5.1 Definition of quasi-redundancy 65

tolerances for both the position of the TCP and the tool orientation there is no bijective

correlation between the TCP- and the MCS movement. The basic configuration is given,

but within the tolerances of the tool orientation there remains an infinite residual diversity

of possible positions of the different axes, which is exemplified in figure 5.1. Therefore in

every machine tool consisting of both rotational and translational axes there exist ad-

ditional degrees of freedom within the tolerance of the tool orientation, which are in the

following denoted as quasi-redundant degrees of freedom.

The tolerance of the tool orientation is defined depending on the manufacturing process.

In the case of flank milling for example only rotations of the tool ∆ϕ around the normal

direction of the manufactured surface n are feasible as shown in figure 5.2. In this case

there exists only one quasi-redundant degree of freedom.

XZ

Y

rT CP (t)

∆ϕB

rMCS(t)

Figure 5.1: Example for quasi-redundancy in the case of a five axis laser cutting machine

tool. In this example a movement of rT CP is enabled by both a translation of rMCS and a

rotation of the B axis within the orientation tolerance ∆ϕB, which leads to compensation

movements of the linear axes.

In the case of five-axis laser cutting and also in the case of end milling the tolerance domain


of the tool orientation can be described in two different ways: In the case of deviations of

the tool orientation, which depend on each other the tolerance domain can be described

as a cone. As already shown in chapter 4 according to Castagnetti [17] and figure 4.3

the more general case with independent admissible deviations of the tool orientation is

described as a pyramid with a rectangular base containing all the possible orientations of

the tool axis for each point on the tool path. Both approaches have in common, that the

admissible deviations of the tool orientation allow for two additional degrees of freedom.

In the following the exploitation of the quasi-redundancy in the case of five axis laser

cutting respectively end-milling is shown at the example of the kinematics of the 5-axis

machine tool shown in figure 3.2.

XZ

Y

n

∆ϕ

workpiece

tool

Figure 5.2: Definition of the tolerance of the tool orientation depending on the manu-

facturing process for flank milling as an example. The rotation ∆ϕ represents the ad-

ditional degree of freedom.

5.2 Conception of the optimisation

As already discussed in chapter 2 the global task for the trajectory optimisation is an

increase of the machining productivity, which is influenced by both the accuracy and the

machining time. For simplicity, the machining time and therefore rT CP (t) is assumed to be

5.2 Conception of the optimisation 67

fixed in the remainder of this chapter since it is evaluated in a previous step. Taking into

account that rMCS(t) denotes the movement of the axes of the machine tool, the accuracy

of the trajectory respectively the mechanical excitation of the machine tool and therefore

the dynamic errors during machining are influenced by an optimisation of rMCS(t). A first

approach in this field is presented by Sellmann et al. [74]. For the example of a 5-axis

laser cutting machine tool the smoothing of the movement of the axes of the machine

tool rMCS(t) is shown. Within the proposed objective function the rate of the curvature of

the trajectory in the machine coordinate system is minimised. The trajectory is described

by quintic polynomials, which are connected C2-continuously. By the use of a rigid-body

model it is shown, that the proposed method leads to reduced mechanical excitations of

the machine tool.

According to Steinlin [79] the mechanical excitation of machine tools due to the trajectory

generation is caused by the following two phenomena: In a frequency range below the

first eigenmode proportional to the acceleration of the trajectory x(t) according to (3.21)

elastic machine parts like guides are deformed quasi static by eccentric actuator forces.

The more relevant phenomenon for the set point generation is an excitation of machine tool

structures in the range of and above the first eigenfrequency, which is primary caused by

the rate of change of the actuator force and is addressed in the remainder of this chapter.

Non regarding frictional effects, the rate of change of the force is proportional to the jerk.

Additionally the inertias of the axes of the machine tool need to be taken into account

during optimisation.

This leads to the following demands on the optimisation method:

• Minimising the jerk values of the different axes for a given rT CP (t)

• Consideration of the inertias of the axes of the machine tool in order to obtain a

prioritisation of these axes during optimisation.

• Compliance with the given tolerance of the tool orientation

Generally speaking during the optimisation with respect to the axes inertias the jerk values...r

MCS(t) are minimised while the given...r

T CP (t) is maintained while exploiting the inner

degrees of freedom.

The physical limitations of a machine tool are chosen in order to ensure a reliable process

and the specified accuracy. Among other machining errors, which are already mentioned

in table 2.1, the accuracy is decreased by mechanical excitation, which is influenced by

the set point generation.


Comparing an acceleration-limited trajectory with a trajectory, that is additionally jerk-

limited, the process time obviously increases for the jerk-limited trajectory, which is already

shown by Steinlin [79]. To avoid this effect the acceleration limitation has to be raised.

Taking into account that the acceleration limitations are set with respect to the possi-

ble actuator force or to reduce dynamic effects like cross-talk, raising of the acceleration

limitations is usually not possible.

Therefore together with a jerk-minimal trajectory of the axes of the machine tool the

jerk-limitations can be raised in order to save process time and to ensure the requested

accuracy, which leads to an increased productivity.

5.3 Problem formulation

As mentioned before, for the exploitation of quasi-redundancy for a given trajectory

rT CP (t), a jerk-minimal trajectory rMCS(t) within the given constraints with respect to

the axes inertia is sought. Therefore in a first step based on the DGO, which was pre-

sented in section 4.3, rT CP (t) is obtained either by the virtual CNC, which was introduced

in section 4.4.1, or is assumed to have a constant feed rate for simplicity. However this first

step leads to a discretised smooth trajectory rT CP (t) and the corresponding time vector

respectively the time steps hi. Because of this all derivatives with respect to the time t

become a linear function of time.

5.3.1 Discretisation

In the following the dynamics of the regarded trajectory r(t) including the movement of

all axes of the machine tool are discretised as

yi

= [ri, ri, ri] (5.1)

for the ith discretisation step. For n discretisation steps the variable space x is defined in

dependency of all states yi

as

x =[y1, . . . , yi

, . . . , yn

](5.2)

System dynamics: According to Steinlin [79] for the dynamics of the regarded trajec-

tory r(t) first order state equations are defined. To satisfy these equations a set of equality

5.3 Problem formulation 69

constraints

ζi = ri+1 − ri −hi

2 (ri+1 + ri) (5.3)

ζi+1 = ri+1 − ri −hi

2 (ri+1 + ri) (5.4)

is defined for each discretisation step and can be assembled to a system of linear equations

Aeqx = beq (5.5)

With (5.5) system dynamics up to the second derivative with respect to time are consid-

ered in the sense of C2-continuity, which is defined and requested in chapter 3.6. Higher

derivatives can be incorporated by additional state equations.

5.3.2 Geometric Constraints

Assuming a fixed rT CP (t) the orientation of the tool along the tool path is ensured during

optimisation by the geometric constraints. The regarded process of 3D laser cutting leads

to the following three options for the constraint formulation.

Individual limitation of the angles of the rotational axes: A basic constraint

formulation is the limitation of the rotation angles B(t) and C(t), which can be obtained

by linear constraints. For the ith discretisation step a feasible tool orientation is obtained

if Bi and Ci comply with

Bi −∆ϕB ≤Bi ≤ Bi + ∆ϕB (5.6)

Ci −∆ϕC ≤Ci ≤ Ci + ∆ϕC (5.7)

In (5.6) and (5.7) Bi and Ci represent the discretised programed angles B respectively C.

∆ϕB respectively ∆ϕC denote the point wise span of the tolerances for each axis.

Isotropic limitation of the tool orientation: In order to allow the tool for rotations

around the tool axis, an isotropic limitation of the tool orientation is requested. As shown

in figure 5.3, all orientations of the tool lying inside of the sketched cone with the opening

angle ∆ϕ are feasible. This is obtained by the limitation of the angle ϕ(ta) between the

actual tool orientation n(ta) and the programed tool orientation n(ta) with


ϕ(ta) = arccos(

n(ta) · n(ta)|n(ta)| · |n(ta)|

)≤ ∆ϕ (5.8)

for every time step ta.

Anisotropic limitation of the tool orientation: In order to consider an anisotropic

limitation of the tool orientation in the case of 3D laser cutting it must be distinguished

between the lead and tilt angle ϕ(t)lead respectively ϕ(t)tilt, which are already mentioned

in section 4.2.2. For this purpose the local coordinate system ψ(t) = {eu(t), ev(t), ew(t)},which was introduced in section 4.2.2, is evaluated. The tilt angle ϕ(t)tilt is defined as

the angle between the tool and ew(t) in the plane, which is spanned by ev(t) and ew(t).Thus the lead angle ϕ(t)lead is defined as the angle between the tool and ew(t) in the plane,

which is spanned by the vectors eu(t) and ew(t).

For the example of 3D laser cutting the lead angle ϕ(t)lead is only limited by the process

tolerance. Depending on the material of the sheet metal as well as the used laser technology

the cutting process only works within a delimited angular deviation between the normal

direction of the sheet metal and the direction of the laser beam. The deviations of the

orientation in positive or negative direction of the cutting edge are negligible for the quality

of the cutting edge. For the tilt angle ϕ(t)tilt on the other hand the maximum skewing of

XZ

Y

∆ϕ

n(ta) n(ta)

rT CP (t)

Figure 5.3: Isotropic Limitation of the tool orientation

5.4 Nonlinear programing approach 71

the cutting edges is the limiting tolerance value, so in this case the process tolerance is

negligible.

Although both the isotropic- and the conical limitation of the tool orientation represent

more general cases for the constraint formulation, from a practical point of view e.g. com-

putational power for the optimisation the individual limitation of the angles B respectively

C are assembled to the constraints

Ax = b, (5.9)

which are first described and are used in the following.

5.3.3 Boundaries

In order to integrate the exploitation of quasi-redundancy in a windowing solution com-

pared to the method described in section 4.3.6, boundary conditions have to be formulated

for the optimisation. This is achieved by a set of linear equality constraints

dkr

dtk(t) = dkr

dtk(t), t = 0, te (5.10)

for the kth derivative of r(t) with respect to time at t = 0 respectively t = te, which are

attached to the linear equality constraints (5.5).

5.4 Nonlinear programing approach

For the general case of large tolerances of the tool orientation a nonlinear programing

(NLP) approach for the exploitation of the quasi-redundancy is developed in the following

for the example of the kinematics of the 5-axis laser cutting machine tool, which is shown

in figure 3.2.

With respect to the constraints and boundary conditions described above for a given tool

path rT CP (t) with 0 ≤ t ≤ te a jerk-minimal curve rMCS(t) as a function of the trajectory

of the rotational axes B(t) respectively C(t) is sought.



The objective function consists of the integral of the square of the jerk values of the

machine tool axes with respect to their different inertias. For n axes this leads to

J =i=n∑

i=1

∫ te

ta

νi(...r

MCS(t)i)2dt. (5.11)

...r

MCS(t)i is the jerk and contains as components the jerk of the different axes. νi are

additional weights, which are chosen proportional to the inertias of the linear axes. For the

example of the regarded 5-axis laser cutting machine tool with the offsets ∆B respectively

∆C, 3-times differentiation of the transformation (3.6) with respect to time leads to

...r

MCS(t)X = sinC(t)(− 3∆CC(t)C(t)− 3∆B cosB(t)

(C(t)B(t)

+ B(t)C(t))

+ ∆B sinB(t)(

3B(t)2C(t) + C(t)3

− ...C(t)

))+ cosC(t)

(− 3∆B sinB(t)

(B(t)B(t)

+ C(t)C(t))

+ ∆B cosB(t)(− B(t)3 − 3B(t)C(t)2 +

...B(t)

)

+ ∆C(− C(t)3 +

...C(t)

))+

...XT CP (t) (5.12)

...r

MCS(t)Y = sinC(t)(− 3∆B sinB(t)

(B(t)B(t) + C(t)C(t)

)

+ ∆B cosB(t)(− B(t)3 − 3B(t)C(t)2 +

...B(t)

)

+ ∆C(− C(t)3 +

...C(t)

))+ cosC(t)

(3∆CC(t)C(t)

+ 3∆B cosB(t)(C(t)B(t) + B(t)C(t)

)

+ ∆B sinB(t)(− 3B(t)2C(t)− C(t)3 +

...C(t)

))+

...Y T CP (t) (5.13)

...r

MCS(t)Z =− 3∆B · cosB(t) · B(t)B(t)

+ ∆B · sinB(t)(B(t)3 − ...

B(t))

+...ZT CP (t) (5.14)

...r

MCS(t)B =...B(t)T CP (5.15)

...r

MCS(t)C =...C(t)T CP (5.16)

Since...XT CP (t),

...Y T CP (t) and

...ZT CP (t) are evaluated in a previous step, all the different

contributors of the objective function only depend on the trajectories of the rotational

axes B and C.


5.4.2 Numerical optimisation

The optimisation of the problem described above is carried out using a standard solver.

In order to feed this solver the optimisation problem described above, which is in fact

nonlinear must be formulated as a nonlinear program (NLP) according to Betts [8]. For a

given variable x ∈ Rn a best solution x∗ is evaluated, which minimises a scalar objective

function

J = f(x) (5.17)

subject to the nonlinear constraints

c(x) ≤ 0 (5.18)

ceq(x) = 0 (5.19)

and the linear constraints

A · x ≤ b (5.20)

Aeq· x = beq. (5.21)

(5.18) and (5.19) are functions that return vectors. In (5.20) and (5.21) b and beq are

vectors, A and Aeq

are matrices. The numerical optimisation is carried out using an

interior-point algorithm, which finds a minimum of constrained nonlinear multivariable

function. The interior-point algorithm allows for providing analytically calculated gradi-

ents for both the objective function and the nonlinear constraints as well as their Hessian.

If the gradient values and the Hessian are not supplied to the algorithm, the computation

time increases and the quality of the obtained solution decreases.

Identification of optimality: Due to the nonlinear problem formulation, which is ad-

ditionally not convex, the optimisation algorithm is not able to find the global minimum

of the objective function. Therefore, an initial guess close to the solution improves the

quality of the optimised solution.

5.4.3 Application example

In the following an application example is shown and discussed in order to demonstrate

the influence of the quasi-redundancy on the machining productivity.


The step up geometry shown in figure 5.4 is a basic example for the quasi-redundancy.

The tool orientation is changed by 90◦ along the TCP-trajectory rT CP in the XZ-plane of

the workspace as shown in figure 5.4. This leads to the compensation movements of rMCS,

which are also shown in figure 5.4. Critical points are the reversal points of rMCS since in

this point at least one axis has to change its direction. At this point a high acceleration

respectively high jerks of the X respectively Z axis and an increased dynamic error is

expected.

Due to the fact that two linear axes and one rotational axis are involved in the trajectory

there is only one quasi-redundant degree of freedom within the given orientation tolerance

of 20◦ for the tilt-angle ∆ϕtilt. Therefore in this example the result is not affected by the

additional weights νi. In addition the objective function (5.11) is simplified assuming

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

300

350

400

450

Position X [mm]

PositionY

[mm

]

rT CP

rMCS

rMCS

Figure 5.4: The TCP-trajectory (black) in the XZ-plane and compensation movements

of the linear axes due to the changed tool orientation (red and green). The exploitation of

the quasi-redundancy along the whole tool path leads to a local smoothing of the reversal

points of the original geometry (red). The direction of the movements of both rT CP and

rMCS are labeled with black arrows.


0 50 100 150 200 250 300 350 400 450−100

−80

−60

−40

−20

0

TCP-path [mm]

PositionB

[◦ ]

rb

rb

Figure 5.5: Original (red) and optimised (green) trajectory of the B-axis which leads to

the compensation movements which are shown in figure 5.4.

C(t) = 0 ∀ t ∈ I := [0, te]. (5.22)

Assuming a constant feed rate for rT CP (t) and non regarding the dynamic limitations of the

machine tool axes, a minimisation of the objective function (5.11) leads to the optimised

movement of the B axis according to figure 5.5 and therefore an optimised trajectory rMCS.

The resulting discretised trajectory is approximated by a B-Spline using the DGO and is

converted to NC-Code, which is given to the virtual CNC in order to obtain a time discrete

feed rate profile. The results for the step up geometry, which are obtained by the virtual

CNC, are shown in figure 5.6-5.8 and are discussed in the following.

As shown in figure 5.6 due to the reduced arc length of rMCS(t), while the maximum feed

rate of rT CP (t) is reached as shown in figure 5.8, the maximum velocity of both the X and

the Z axis can significantly be reduced.

Regarding figure 5.7 the global smoothing of the given trajectory leads to a continuous

acceleration of the different axes. Additionally the maximum acceleration in the vicinity

of the transitions is reduced.

As shown in figure 5.8 velocity collapses along the tool-path are significantly reduced,

which leads to a time saving of about 20%. This is caused by both the decreased arc

length and the smoothness of rMCS(t) mentioned above. Therefore the trajectory is only

limited by the maximum feed rate at the TCP. The other dynamic limitations of the

involved axes are not affected.


0 50 100 150 200 250 300 350 400 450−1,000

−500

0

500

1,000

1,500

TCP-path [mm]

Velocity

[mm

/s]

XMCS

˜XMCS

ZMCS

˜ZMCS

Figure 5.6: Resulting velocity of the X-axis (red) and the Z-axis(blue) for both the

original rMCS (solid line) and the optimised trajectory rMCS (dashed line).

Application limitations: Although the used solver allows for the analytical calculation

of the gradient respectively Hessian of the objective-function and the nonlinear constraints

the calculation of these additional arguments is very extensive. Especially the calculation

of the Hessian of the objective-function takes very much computational power, which limits

the number of discretisation steps. The inclusion of the physical limitations of the different

axes by nonlinear constraints would additionally complicate the regarded problem due to

the fact, that for all the physical limitations derivatives of (3.6) with respect to time t

need to be taken into account as well as their gradients and their Hessian. Therefore a

simplification of the NLP is required, which is figured out in the following.

5.5 Quadratic programing approach

The achievement of the quadratic programing approach (QP), which is described in the

following is comparable to the NLP described above. Based on a discretised trajectory

rT CP (t) a jerk-minimal trajectory rMCS(t) is sought within the given geometric and dyn-

amic constraints with respect to the inertia of the axes of the machine tool. First of all

the proposed linearisation of the NLP is pointed out. Afterwards the description of the

dynamic constraints of the different axes as well as the objective function for the quadratic

programing approach are presented.

5.5 Quadratic programing approach 77

0 50 100 150 200 250 300 350 400 450

−2

−1

0

1

2

TCP-path [mm]

Acceleration

[104

mm

/s2 ]

XMCS

˜XMCS

ZMCS

˜ZMCS

Figure 5.7: Resulting acceleration of the X-axis (red) and the Z-axis(blue) for both the

original rMCS (solid line) and the optimised trajectory rMCS (dashed line). The global

smoothing leads not only to a continuous acceleration but also to a reduced average accel-

eration for both the X and the Z-axis.

5.5.1 Linearisation

As shown in the previous example in the general case for the example of 5-axis laser cutting

a large orientation tolerance can be assumed. Although quasi-redundancy is available along

the whole trajectory, the main advantage can be gained by optimisation of trajectories with

large changes of the tool orientation along a short arc length of rT CP (t) and therefore large

compensation movements of the linear axes. With a programed tool orientation collinear to

the normal vector n to the sheet metal, these trajectories arise in areas of large curvatures

of the sheet metal, where the orientation tolerance must be reduced to ∆ϕ ≤ ±5◦ in order

to avoid collisions. Assuming a limitation of the tolerance of the rotational axes of ±5◦ this

offers an opportunity for a linearisation of the quasi-redundancy on the one hand. On the

other hand, a rotation around the tool-axis is not possible, which is advantageous for the

avoidance of collisions, but leads to a restriction of the solution space of the optimisation

algorithm for the exploitation of the quasi-redundancy.

Assuming that the orientation tolerance for the exploitation of the quasi-redundancy is

limited to a small angle δB respectively δC the ith value of the optimised trajectories B(t)respectively C(t) can be approximated as

Bi = Bi + δBi (5.23)

Ci = Ci + δCi (5.24)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

100

200

300∆t = 20%

Time [s]

Feedrate

[mm

/s]

Original geometry rT CP (t)Optimised geometry rT CP (t)

Figure 5.8: Resulting feed rate at the TCP. Due to the resolution of the quasi-redundancy

the velocity collapses along the tool-path vanish, which leads to a time saving of 20%.

with the ith given orientation Bi respectively Ci and

−5◦ ≤ δBi ≤ 5◦ (5.25)

−5◦ ≤ δCi ≤ 5◦ (5.26)

for i = 1, . . . , n.

Using the addition theorems according to Bronstein [44] with (5.23) and (5.24) it is

sin(Bi) = sin(Bi + δBi) = sin(Bi)cos(δBi) + cos(Bi)sin(δBi) (5.27)

sin(Ci) = sin(Ci + δCi) = sin(Ci)cos(δCi) + cos(Ci)sin(δCi) (5.28)

cos(Bi) = cos(Bi + δBi) = cos(Bi)cos(δBi)− sin(Bi)sin(δBi) (5.29)

cos(Ci) = cos(Ci + δCi) = cos(Ci)cos(δCi)− sin(Ci)sin(δCi). (5.30)

(5.27)-(5.30) in (3.6) leads to

rMCS,i,X = rT CP,i,X + ∆B((sin(Bi)cos(δBi) + cos(Bi)sin(δBi)

)

(cos(Ci)cos(δCi)− sin(Ci)sin(δCi)

))

+∆C(sin(Ci)cos(δCi) + cos(Ci)sin(δCi)

)(5.31)

rMCS,i,Y = rT CP,i,Y + ∆B((sin(Bi)cos(δBi) + cos(Bi)sin(δBi)

)


(sin(Ci)cos(δCi) + cos(Ci)sin(δCi)

))

+∆C(cos(Ci)cos(δCi)− sin(Ci)sin(δCi)

)(5.32)

rMCS,i,Z = rT CP,i,Z + ∆B((cos(Bi)cos(δBi)− sin(Bi)sin(δBi)

))(5.33)

Assuming that

cos(δBi) ≈ 1 (5.34)

cos(δCi) ≈ 1 (5.35)

sin(δBi) ≈ δBi (5.36)

sin(δCi) ≈ δCi (5.37)

sin(δBi)sin(δCi) ≈ 0 (5.38)

leads to

rMCS,i,X = rT CP,i,X + ∆Bsin(Bi)cos(Ci) + ∆Csin(Ci)

+ δBi

(∆Bcos(Bi)cos(Ci)

)

+ δCi

(∆Ccos(Ci)−∆Bsin(Bi)sin(Ci)

)(5.39)

rMCS,i,Y = rT CP,i,Y + ∆Bsin(Bi)sin(Ci)−∆Ccos(Ci)

+ δBi

(∆Bcos(Bi)sin(Ci)

)

+ δCi

(∆Csin(Ci) + ∆Bsin(Bi)cos(Ci)

)(5.40)

rMCS,i,Z = rT CP,i,Z + ∆Bcos(Bi)− δBi∆Bsin(Bi). (5.41)

Together with the offsets ∆B respectively ∆C (5.34)-(5.38) lead to an error of the resulting

trajectory rMCS(t) and their derivatives with respect to time t. Due to the fact that

admissible deviations of the tool orientation are directly limited by (5.25) and (5.26) the

fulfillment of the geometric constraints is not affected. Therefore during optimisation a

dynamic reserve must be ensured. For simplicity this is done in the following by scaling

of the time vector.

(5.39)-(5.41) are linear dependent on δB(t) respectively δC(t), so (3.6) is sufficiently ap-

proximated and can be attached to the linear equality constraints (5.5). With the state


equations (5.3) and (5.4) and the given time vector t, both the velocity rMCS,i and the ac-

celeration rMCS,i can also be established as additional states during optimisation by linear

equality constraints.

5.5.2 Dynamic Constraints

Due to the additional state variables rMCS,i and rMCS,i with the physical limitations Φmax

and Φmin the limitation of the kth derivative with respect to time of the different axes at

the ith discretisation step can be incorporated according to Steinlin [79] by the normalised

linear constraints

dkrMCS,i

dtk/Φmax ≤ 1 Φmax > 0 (5.42)

dkrMCS,i

dtk/Φmin ≤ 1 Φmin < 0, (5.43)

which are attached to the inequality constraints (5.9). A jerk limitation is not consid-

ered, since within the optimisation the integral of the square of the jerk for every axis is

minimised using the objective function described in the following.


With respect to Steinlin [79] the jerk...r

MCS,i is not defined by the state equations so the

jerk value for the objective function is determined by first order finite differences of the

acceleration values rMCS,i

...r

MCS,i =rMCS,i+1 − rMCS,i

hi

. (5.44)

With (5.44) (5.11) can be rewritten as

J = 12x

THx. (5.45)

Together with the linear constraints for the state equations, the linearised transformation,

the dynamic constraints and the boundaries this leads to a quadratic program with linear

constraints, which can be solved in the same way as the DGO shown in section 4.3.


5.5.4 Influence of the additional weights

The global task of the exploitation of quasi-redundancy is to reduce the dynamic error due

to the mechanical excitation of the machine tool caused by high changes of the actuator

forces. These are proportional to the jerk as well as the inertia of the axes of the machine

tool.

Both the nonlinear and the linearised optimisation approach described above obtain a

simultaneous solution for all axes of the machine tool. This leads to the opportunity of

a prioritisation of the different axes. Especially in the case of a machine tool with strong

differing inertias of the axes the finally resulting changes of the actuator forces can be

reduced by the prioritisation of the different axes.

Therefore in order to consider the inertias of the different axes the additional weights

νi are used in the linear combination of (5.11). Their influence during optimisation is

demonstrated by a basic test geometry, which consists of a linear positioning of the X,

Y and the C axis by 180mm respectively 180◦. The offset of the C axis ∆C, shown in

figure 3.2 is 163mm. This leads to the trajectory for rT CP (t) respectively rMCS(t), which

are shown in the figures 5.9 and 5.10.

For simplicity and because of the fact that the X respectively the Y axis of the machine

tool shown in figure 3.2 have the major inertias, only the influence of νx respectively νy

is regarded in the following. In order to demonstrate their influence the optimisation is

executed using a vector νi = {νX , νY , νZ , νB, νC} with the following three tuples

0 0.5 1 1.5 2 2.5 3 3.5 4−200

0

200

400

Time [s]

Position

[mm

,◦]

XMCS

Y MCS

ZMCS

BMCS

CMCS

Figure 5.9: Trajectory of the machine axes due to a linear positioning of 180mm of the

X and Y -axis combined with a linear positioning of the C-axis of 180◦


0100

200300

−200−100

0100

200300

4000

100

200

Position X [mm]

Position Y [mm]

PositionZ

[mm

]rT CP

rMCS

Figure 5.10: Resulting trajectories for rT CP (t) and rMCS(t) due to the positioning of the

X, Y and the C axis described in figure 5.9

ν1 = {1, 1, 1, 1, 1} (5.46)

ν2 = {10, 1, 1, 1, 1} (5.47)

ν3 = {1, 10, 1, 1, 1} (5.48)

Figure 5.11 reveals, that the global optimisation of the tool orientation with a tolerance

range of ±5◦ leads to a deviation of the different solutions for rMCS(t) up to 25mm because

of the rearrangement of the actuator forces due to the trajectories of the rotational axes.

The general use case of a 5 axis laser cutting machine tool is the cutting of free-form sheet

metal with a usually high local curvature since it was moulded in a previous step. This

leads to a local large movement of the B and C axis along a short arc length and therefore

large compensation movements, so the orientation tolerance must be chosen very carefully.

Otherwise an optimisation leads to a collision of the machine tool with the sheet-metal.

Due to the fact that the euclidean length of the MCS-trajectory rMCS is changed during

optimisation for a given duration of the TCP-trajectory rT CP known from the virtual CNC

the velocity of the different axes also changes according to figure 5.12.


0 0.5 1 1.5 2 2.5 3 3.5 4−200

0

200

400

Time [s]

Position

[mm

,◦]

XMCS

XMCS,ν1

XMCS,ν2

XMCS,ν3

Y MCS

YMCS,ν1

YMCS,ν2

YMCS,ν3

CMCS

CMCS,ν1

CMCS,ν2

CMCS,ν3

Figure 5.11: Resulting position of the machine axes due to the optimisation. Depending

on the tuple of additional weights νi according to (5.46)-(5.48) the exploitation of the

quasi-redundancy leads to a deviation of the position up to 25mm in the machine coordinate

system.

0 0.5 1 1.5 2 2.5 3 3.5 4−100

0

100

200

300

Time [s]

Velocity

[mm

/s,◦

/s]

XMCS

XMCS,ν1

XMCS,ν2

XMCS,ν3

Y MCS

YMCS,ν1

YMCS,ν2

YMCS,ν3

CMCS

CMCS,ν1

CMCS,ν2

CMCS,ν3

Figure 5.12: Resulting velocity of the machine axes due to the optimisation with the tuple

of additional weights νi according to (5.46)-(5.48).


Figure 5.13 respectively 5.14 illustrate the intended rearrangement of the actuator accel-

eration and their derivative with respect to time. Obviously an optimisation with ν1 leads

to a first reduction of the maximum jerk of all axes. As required for ν2 respectively ν3

both the maximum acceleration and the jerk of the X and the Y axis are rearranged.

This leads to decreased maximum acceleration respectively jerk for the X axis for ν2 and

a decreased maximum acceleration respectively jerk for the Y axis for ν3

In the remainder of this chapter ν is chosen proportional to the axes inertias known from

the acceptance test record of the machine tool. Since νi is included in the objective function

(5.11) not the absolute value of the inertias but the relationship between the different

inertias is important for the optimisation. This leads to νi = {0.3, 0.1, 0.07, 0.01, 0.01}.

5.6 Application Example

In the following section the exploitation of the quasi-redundancy is demonstrated for the

example of planar geometry in the XY plane of the workspace using an orientation toler-

ance of ±5◦. While the previous example only consists of a straight positioning movement

the geometry shown in figure 5.15 has sharp corners. As shown in figure 5.16 these are

optimised in a previous step using the DGO, which is described in 4.3. The time-discrete

0 0.5 1 1.5 2 2.5 3 3.5 4

−200

0

200

400

Time [s]

Acceleration

[mm

/s2 ,

◦/s

2 ]

XMCS

XMCS,ν1

XMCS,ν2

XMCS,ν3

Y MCS

YMCS,ν1

YMCS,ν2

YMCS,ν3

CMCS

CMCS,ν1

CMCS,ν2

CMCS,ν3

Figure 5.13: Resulting acceleration of the machine axes due to the optimisation with the

tuple of additional weights νi according to (5.46)-(5.48).

5.6 Application Example 85

0 0.5 1 1.5 2 2.5 3 3.5 4−2,000

0

2,000

4,000

6,000

Time [s]

Jerk

[mm

/s3 ,

◦/s

3 ]

XMCS

XMCS,ν1

XMCS,ν2

XMCS,ν3

Y MCS

YMCS,ν1

YMCS,ν2

YMCS,ν3

CMCS

CMCS,ν1

CMCS,ν2

CMCS,ν3

Figure 5.14: Resulting jerk of the machine axes due to the optimisation with the tuple

of additional weights νi according to (5.46)-(5.48).

feed rate profile is obtained using the virtual CNC. The trajectories of the rotational axes

are optimised using the quadratic programing approach which is described in section 5.5.

The optimisation leads to the trajectories for the B respectively C axis according to

figure 5.17. Obviously the given orientation tolerance is completely exploited for the B-

axis. The orientation tolerance of the C-axis is not exploited, since in this example this

axis is primary used for an infeed movement, which leads to quasi-redundancy between

the B and the X-axis.

Figure 5.18 shows the position of the X, Y and Z axis for both the original and the

optimised geometry. Except for the compensation movements of the X-axis the Y and

the Z axis are hardly affected by the optimisation since for this example the initial angle

of the C axis leads to a configuration of the kinematic, which primarily allows for a quasi-

redundancy in the X-direction as mentioned above.

The resulting trajectories rMCS(t) and rMCS(t) of the original and the optimised geometry

are shown in figure 5.19.

Regarding figure 5.20-5.22 the maximum velocity, acceleration as well as the jerk of the X-

axis are significantly reduced by the exploitation of the quasi-redundancy. As mentioned

before the Y -axis is not affected by the optimisation. The compensation movement due

to the trajectory of the B and C-axis shown in figure 5.17 is performed by the Z-axis.


−20 −15 −10 −5 0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

45

Position X [mm]

PositionY

[mm

]rT CP

rT CP

Figure 5.15: Planar geometry in the XY -plane, which is optimised in a previous step

using the DGO. A magnification of the right corner is shown in figure 5.16.

In order to compare the resulting mechanical excitation the spatial deviation between

the internal measurement system and rMCS(t) respectively rMCS(t) is simulated with the

machine model, which is introduced in section 4.4.2 as shown in figure 5.23. Taking into

mind the significant reduction of the jerk of the machine tool axes due to the optimisation,

the reduction of the spatial deviation shown in figure 5.23 is comparatively small. An

evaluation of the deviation at the TCP would be more sufficient since the compliance

of the whole machine structure is considered during simulation. Because of the strong

differing dynamics of the machine tool axes the contouring errors of the different axes

strongly differ, which leads to a significant deviation at the TCP for the machine model.

For the synchronisation of the contouring error of the linear axes circularity test can be

used. With the help of these tests the contouring error can also be reduced for higher

dynamics of the machine tool axes. For the synchronisation between rotary and linear

axes a suitable method such as the circularity test is not available, so an optimal setting

of the control parameters of the axes is not directly available. Further studies should be

5.6 Application Example 87

26 28 30 32 34

26

28

30

32

34

Position X [mm]

PositionY

[mm

]

rT CP

rT CP

Figure 5.16: Magnification of the right corner of figure 5.15 in order to show the opti-

misation of r(s).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−4

−2

0

2

4

t [s]

Angle

[◦ ] BMCS

BMCS

CMCS

CMCS

Figure 5.17: Resulting trajectory due to the optimisation for the B and C axis with

respect to time.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−200

−100

0

100

200

t [s]

Position

[mm

] XMCS

XMCS

Y MCS

YMCS

ZMCS

ZMCS

Figure 5.18: Resulting trajectory rMCS(t) for the X, Y and Z for both the original and

the optimised geometry.

focussed on these methods.

5.7 Discussion

For the exploitation of the quasi-redundancy two optimisation approaches are described

and exemplified above. The nonlinear optimisation approach, which is presented in sec-

tion 5.4 allows for an optimisation in the general case of a large orientation tolerance. In

order not to suffer from the extensive evaluation of both the gradient and the Hessian of

the nonlinear optimisation approach a linearisation for a small orientation tolerance is per-

formed, which leads to the quadratic programing approach described in section 5.5. Just

as the DGO, which is described in section 4.3 the arising quadratic optimisation problem is

convex and therefore allows for a reformulation as a second order cone program according

to Boyd [14], which offers the capability of a real time application.

The limitation of the tolerance span and therefore the choice of one of the described

optimisation approaches must be figured out in response to the regarded trajectory. So

while in the general case in every point along the tool path the whole span of the quasi-

redundant degrees of freedom are available, the quadratic programing approach in the

worst case only allows for the optimisation in the direction of one axis as shown in the

application example. A basic opportunity to overcome this disadvantage is a preprocessing

of the initial angles of the B- and the C- axis within the given constraints in a previous

step.

5.7 Discussion 89

−20 −15 −10 −5 0 5 10 15 20 25 30 35 40

−160

−155

−150

−145

−140

−135

−130

−125

−120

−115

X [mm]

Y[m

m]

rMCS

rMCS

Figure 5.19: rMCS(t) of the original and the optimised geometry. Due to the primarily

optimised trajectory of the X-axis the span of the geometry in the Y -direction is hardly

affected.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−200

−100

0

100

200

t [s]

Velocity

[mm

/s] XMCS

˜XMCS

Y MCS

˜Y MCS

ZMCS

˜ZMCS

Figure 5.20: Resulting velocity of the X-, Y - and Z-axis.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−5,000

0

5,000

t [s]

Acceleration[m

m/s

2 ] XMCS

˜XMCS

Y MCS

˜Y MCS

ZMCS

˜ZMCS

Figure 5.21: Resulting acceleration of the X-, Y - and Z-axis.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−5

0

5

10

t [s]

Jerk[1

05m

m/s

3 ]

...XMCS

...XMCS...Y MCS

...Y MCS...Z MCS

...Z MCS

Figure 5.22: Resulting jerk of the X-, Y - and Z-axis.

5.7 Discussion 91

0 20 40 60 80 100 120 1400

10

20

30

40

50

Path [mm]

Deviation

[µm

]Original geometry rMCS(t)Optimised geometry rMCS(t)

Figure 5.23: Spatial deviation between rMCS(t) respectively rMCS(t) and the internal mea-

surement system.

92 6. Tolerance model

Chapter 6

Tolerance model

In the following a tolerance model for the derivation of the position dependent available

smoothing tolerance is presented. The motivation for the investigation of a tolerance

model is explained in section 6.1. The general procedure for the calculation of the avail-

able smoothing tolerance based on geometric uncertainties of a machine tool is shown

in section 6.2. At the examples of 3- and 5-axis machine tools the computation of the

geometrical uncertainties are demonstrated in the sections 6.3 and 6.4.

6.1 Motivation

The smoothing of the tool path of state of the art NC controllers is usually performed

assuming a constant value for the smoothing tolerance. According to Sellmann et al. [75]

this value has to be smaller than the final tolerance value, which is obtained for the finished

part because of additional errors during the manufacturing process. These errors consist of

geometric, dynamic and thermal errors as well as the effect of follow-up errors of the axis

drives pursuant to Andolfatto et al. [4]. All these errors depend on different parameters,

which can vary during machining. In order to have only one tolerance value, the smoothing

algorithm has to deal with the worst case for the above described dependency for the whole

workspace. As a consequence of that the resulting value for the smoothing tolerance is

very small in order to guarantee the requested accuracy. This leads to higher curvatures of

the tool path and finally to a higher mechanical excitation of the machine tool for a given

path velocity. Figure 6.1 illustrates the effect of different path tolerances on the resulting

curvature for corners having an angular range between 0 (straight) and 120 degrees.

Obviously, for a given corner angle a smaller smoothing tolerance ∆n increases the resulting

curvature and vice versa. Assuming that the forces perpendicular to the tool path and

6.1 Motivation 93

10 20 30 40 50 60 70 80 90 100 110 120

5

10

15

20

25

30

1

2

5

1020

50

100200

Angle [◦]

∆n

[µm

]

0

20

40

60

80

100

120

140

160

180

200

Max

imum

curvature

[1/m

m]

Figure 6.1: Contour plot of the resulting curvatures for a smoothed corner depending

on the smoothing tolerance and the angular range of the corner. The colorbar denotes the

maximum curvature of the smoothing function.

therefore the resulting loads on the machine tool are proportional to the curvature of the

tool path a maximum exploitation of the smoothing tolerance is desirable.

Geometric errors, which are defined in ISO230-1 [46] are the biggest non-thermal sources

of inaccuracy, so the tolerance model, which is described below predicts the position de-

pendent smoothing tolerance for a given workpiece. Based on the geometric errors the

resulting geometric uncertainties in the sensitive direction of the tool path are calculated

in a first step. In a second step the smoothing tolerance is suitably reduced.

Thermal and dynamic errors as well as follow-up errors are not regarded since they occur

during machining, so a prediction for the smoothing of the tool path is not possible.

They are considered by an additional reduction of the available smoothing tolerance by a

constant value.


6.2 General procedure

In the following section the general procedure for the derivation of the position dependent

smoothing tolerance is presented. Definitions used in the course of the remaining chapter

are pointed out in section 6.2.1. The computation of the geometric uncertainties based

on the known geometric errors is shown in section 6.2.2. Finally the overall method is

described in section 6.2.3.

6.2.1 Definitions

For the derivation of the available smoothing tolerance different tolerance values have to

be defined.

The tolerance of the workpiece, which is known from the drawing, in the following denoted

as ∆dwg is shown in figure 6.2.

Assuming a machine tool without any machining errors and a rectangular distribution of

the occurring deviations, the NC tolerance in the following denoted as ∆NC for a given

∆dwg, is calculated as follows.

According to ISO230-9 [47] and figure 6.2 the standard uncertainty udwg due to deviations

up to ∆dwg is defined as

udwg = (L+ ∆dwg)− (L−∆dwg)2√

3= 2∆dwg√

12. (6.1)

L−

∆dw

g

L+

∆dw

g

Figure 6.2: ∆dwg known from the technical drawing.

6.2 General procedure 95

∆NC denotes the possible range of deviations, which lead to a combined uncertainty uNC

comparable to udwg. The standard uncertainties u1,2 due to deviations up to ∆NC are

defined as

u1 = u2 = ∆NC√12

(6.2)

According to ISO 230-9 [47] the combined standard uncertainty uc resulting from u1,2 is

defined as

uNC =√u2

1 + u22 =√

2u1,2 (6.3)

Equalising of (6.1) and (6.3) leads to

∆NC =√

2∆dwg (6.4)

The smoothing tolerance in the following denoted as ∆smt is the NC tolerance reduced by

the uncertainties Ui as a result of the machining errors due to

• thermal deviation

• dynamic deviation

• process deviation

• geometric deviation

as illustrated in figure 6.3

6.2.2 Calculation of geometric uncertainties

The uncertainty of measurement data in general is defined as the distribution of values

attributed to a measured quantity.

In the case of machine tools according to ISO230-9 [47] the uncertainty is defined as follows:

∆NC

∆smt

U U UU

Figure 6.3: Available smoothing tolerance ∆smt which is calculated by the reduction of

∆NC by the uncertainties Ui due to machining errors.


The individual contributors to the measurement uncertainty are identified and expressed

as standard uncertainties ui. The combined standard uncertainty uc is calculated as

uc =√u2

r +∑

u2i (6.5)

with

uc: combined standard uncertainty

ur: sum of strongly positive correlated contributors (6.6)

ui: standard uncertainty of i uncorrelated contributor

ur =∑

uj (6.6)

where uj is the standard uncertainty of j strongly positive correlated contributors.

The measurement uncertainty U is calculated by

U = cv · uc (6.7)

with

U : measurement uncertainty

cv: coverage factor. Usually cv = 2, which defines an interval having a probability of

approximately 95%. For example cv = 3 defines an interval having a probability

greater than 99%.

uc: combined standard uncertainty

A standard uncertainty ui is obtained by statistical analysis of experimental data or by

other means, such as knowledge, experience and scientific guess. If an estimation gives a

possible range of ±a or (a+−a−) of a contributor and assuming a rectangular distribution

the standard uncertainty ui is given by

ui = a+ − a−2√

3(6.8)

with

ui: standard uncertainty

6.2 General procedure 97

a+: the upper limit of the rectangular distribution

a−: the lower limit of the rectangular distribution

For the calculation of the geometric uncertainties the component errors and the location

errors pursuant to ISO230-1 [46] are used. All these contributors are uncorrelated so the

ur in (6.5) vanishes.

The range for the evaluation of the uncertainties is given by measurement data like the

acceptance test of the machine tool. For estimation of the different contributors it should

be considered if the regarded geometric error depends on the position. In the case of a

position dependent error, which is shown in figure 6.4, the geometric error can be defined

as U = U0 + UL0LL0

. For position independent errors, which are shown in figure 6.5, the

geometric error should be defined as U = Utotal.

The calculation of the effective uncertainty depending on the kinematics of the machine

tool and the geometry of the workpiece is presented in the following section. All calcu-

lations are based on the assumption, that the uncertainty of a point in the workspace

remains nearly constant within the tolerance band. This is valid for small tolerances.

6.2.3 Overall description of the method

Pursuant to figure 6.6 the workflow for the calculation of the available smoothing tolerance

∆smt is as described below.

Evaluation of the axe wise movement: Based on ISO230-1 [46] the geometric errors

of a machine tool consist of the axe-wise component errors as well as the location errors

between the different axes. A composition of all these contributors leads to a resulting

U0

U(EXX)

L

U

X − stroke

Figure 6.4: Position dependent error.


U0

U(EXX)

L

Utotal

X − stroke

Figure 6.5: Position independent error.

NC-Code

rMCS(s)

UMCS(s) measurement

dataTransformation

MCS ⇔ TCP

UT CP (s)

ψ(s)

Uueff (s)

∆smt(s) ∆NC∆dwg

process errors

thermal errors

dynamic errors

Evaluation

of the axe wise

movment

Calculation

of the effective

geometric uncertainties

Calculation

of the available

smoothing tolerance

Figure 6.6: Workflow for the calculation of the available smoothing tolerance.

6.3 Geometric uncertainties of a 3-axis machine tool 99

geometric error at the TCP for a given geometry. Based on the span of the resulting error

at the TCP the geometric uncertainty can be evaluated as described above. Since the

resulting geometric errors at the TCP are caused by the geometric errors of the machine

tool axes in a first step based on the programmed geometry the trajectory rMCS(s) is

evaluated.

Calculation of the effective geometric uncertainties: Based on rMCS(s) for the

manufacturing of a given workpiece for every s inside of the closed interval I := [a, b]an uncertainty UT CP = {UX , UY , UZ} is evaluated. Depending on the kinematic of the

machine tool the resulting uncertainties at the tool center point UT CP are calculated using

the transformation rMCS(s)⇔ rT CP (s).

The effective uncertainty Ueff is calculated by projection of UT CP in the sensitive direction

of rT CP . In the case of laser cutting the sensitive direction is congruent to the normal

direction ev(s), which is known from the local psi frame ψ(s).

Calculation of the available smoothing tolerance: The following 3 steps finally

lead to the available smoothing tolerance:

1. Calculation of ∆NC based on the technical drawing and ∆dwg according to (6.4).

2. Calculation of the effective uncertainties based on the geometric errors according to

section 6.2.2.

3. Reduction of ∆NC by constant values for

• thermal deviation

• dynamic deviation

• process deviation

and position dependent values for the effective geometric uncertainties.

6.3 Geometric uncertainties of a 3-axis machine tool

In the following the calculation of the geometric uncertainties of a 3-axis machine tool is

demonstrated and exemplified based on Sellmann et al. [75].


6.3.1 Calculation

As defined and illustrated in ISO 230-1 [46] there are 6 component errors defined for each

linear axis listed for the example of a linear X-axis:

EXX: positional deviation

EYX: linear deviation

EZX: linear deviation

EAX: roll deviation

EBX: pitch deviation

ECX: yaw deviation

For 3 linear axes the 18 component errors and the location errors

C0Y : out of squareness between X and Y

B0Z: out of squareness between X and Z

A0Z: out of squareness between Y and Z

yield to 21 geometric errors for a 3 axis machine. All these error contributors are merged

to the axis wise geometric standard uncertainties UX , UY and UZ . For every position of

the trajectory the geometric uncertainties can be calculated according to

UX = 2√12√EXX2

T CP + EXY 2T CP + EXZ2

T CP + ∆Y 2 · C0Y 2 + ∆Z2 ·B0Z2 (6.9)

UY = 2√12√EYX2

T CP + EY Y 2T CP + EY Z2

T CP + ∆Z2 · A0Z2 (6.10)

UZ = 2√12√EZX2

T CP + EZY 2T CP + EYX2

T CP (6.11)

The contributors subscripted with TCP are the resulting deviation ranges at the TCP,

which are also indicated in the acceptance protocol of the machine tool.

6.4 Geometric uncertainties of a 5-Axis machine tool 101

6.3.2 Example

For the example of 2D laser cutting figure 6.7 shows the effective uncertainty in lateral

direction for a 2D work piece. In this case, for a quite small work piece and when only two

axes are involved in the movement, the resulting uncertainties only slightly differ along the

path and are mostly influenced by the contributors due to the squareness error between

the X and the Y axis.

6.4 Geometric uncertainties of a 5-Axis machine tool

Based on Sellmann et al. [75] the calculation of the geometric uncertainties of a 5-axis

machine tool is demonstrated in the following.

6.4.1 Calculation

Regarding a rotational axis there are 6 component errors defined, listed for the example

of a rotational C-axis with respect to ISO230-1 [46].

895 900 905 910 915 920 925 930 935 940

905

910

915

920

925

930

935

Position X [mm]

Pos

itio

nY

[mm

]

30

30.01

30.01

30.02

30.02

30.03

30.03

30.04

30.04

30.05

30.05

Eff

ecti

veunce

rtai

nty

[µm

]

Figure 6.7: Effective uncertainty in lateral direction for a 2D laser cutting work piece.


EXC: radial error motion

EY C: radial error motion

EZC: axial error motion

EAC: tilt error motion

EBC: tilt error motion

ECC: positional deviation

Together with the geometric errors of the 3 linear axes described in section 6.3, the com-

ponent errors for 2 rotational axes and the additional location errors lead to the geometric

uncertainties for a 5 axis machine tool. Regarding the kinematics shown in figure 3.2

dominant contributors for the calculation of the effective uncertainties are the location

errors X0C, Y 0C, X0B and Z0B as well as the component errors ECC and EBB. The

remaining contributors are assumed to be negligible.

Regarding the positional deviation of the C-axis ECC with the span sECC and the radius

rC the resulting sinusoidal contributors in the X and Y direction of the TCP are

EXCT CP (C) = −(ECC(C) + sECC

2

)· sin(C) · rC (6.12)

EY CT CP (C) =(ECC(C) + sECC

2

)· cos(C) · rC . (6.13)

For the calculation of the effective uncertainty at the TCP due to the positional error

ECC only the span sECC of the C axis during the movement Cpart must be taken into

account. This leads to

UX,ECC = 2rC√12

(max

((ECC (Cpart) + sECC

2

)· sin (Cpart)

)

−min((ECC (Cpart)−

sECC

2

)sin (Cpart)

))(6.14)

UY,ECC = 2rC√12

(max

((ECC (Cpart) + sECC

2

)· cos (Cpart)

)

−min((ECC (Cpart)−

sECC

2

)cos (Cpart)

)). (6.15)

Alternatively the whole span sA of the C axis during the movement can be taken into

account, which leads to


UX,ECC = 2rC√12sA (6.16)

UY,ECC = 2rC√12sA. (6.17)

As a consequence of that the contributors of the effective uncertainty at the TCP due to

the positional error strongly increase for small angles.

The location errors X0C and Y 0C mentioned above are described in polar coordinates by

a radius rC and an angle ϕ0 according to figure 6.8. Together with the programmed angle

Cpart this leads to the resulting deviation at the TCP in the X and Y direction due to

X0C and Y 0C

EX(X0C,Y 0C) = rC · cos(ϕ0 + Cpart) (6.18)

EY(X0C,Y 0C) = rC · sin(ϕ0 + Cpart) (6.19)

Comparable to the evaluation of UX,ECC and UY,ECC mentioned above for the calculation

of the effective uncertainty due to the location errors X0C and Y 0C only the span of the

C axis during the movement must be taken into account. For a given angle C this leads

to

UX,(X0C,Y 0C) = 2rc√12

(max (cos (ϕ0 + Cpart))−min (cos (ϕ0 + Cpart))) (6.20)

UY,(X0C,Y 0C) = 2rc√12

(max (sin (ϕ0 + Cpart))−min (sin (ϕ0 + Cpart))) (6.21)

Regarding the B axis of the kinematics shown in figure 3.2 the current position of the C

axis must be taken into account for the calculation of the effective uncertainty. For the

positional error EBB this leads to

EXBT CP (B) = −(EBB (B) + sEBB

2

)· sin (B) · sin (C) · rB (6.22)

EY BT CP (B) = −(EBB (B) + sEBB

2

)· sin (B) · cos (C) · rB (6.23)

EZBT CP (B) = −(EBB (B) + sEBB

2

)· cos (B) · rB (6.24)

As already mentioned above for the calculation of the effective uncertainty only the span

of the regarded axis during the movement must be taken into account. For the effective

uncertainty of the B axis due to the positional error EBB this leads to


UX,EBB = 2rB√12

(max

((EBB (Bpart) + sEBB

2

)· sin (Bpart)

)

−min((EBB (Bpart)−

sEBB

2

)· sin (Bpart)

))· sin (Cpart) (6.25)

UY,EBB = 2rB√12

(max


2

)· sin (Bpart)

)


sEBB

2

)· sin (Bpart)

))· cos (Cpart) (6.26)

UZ,EBB = 2rB√12

(max


2

)· cos (Bpart)

)


sEBB

2

)· cos (Bpart)

))(6.27)

Comparable to the location errors of the C axis the location errors X0B and Z0B are

considered using an initial angle β0 and a radius rB according to figure 6.8. Together with

the programmed angle ϕB this leads to

EX(X0B,Z0B) = rB · cos (β0 +Bpart) · sin (C) (6.28)

EY(X0B,Z0B) = rB · sin (β0 +Bpart) · cos (C) (6.29)

EZ(X0B,Z0B) = rB · sin (β0 +Bpart) (6.30)

for the resulting deviation at the TCP. Taking into account the span of the B axis during

the movement this leads to

UX,(X0B,Z0B) = 2rB√12

(max (cos (β0 +Bpart) · sin (Cpart))

−min (cos (β0 +Bpart) · sin (C)))

(6.31)

UY,(X0B,Z0B) = 2rB√12

(max (sin (β0 +Bpart) · cos (Cpart))

−min (sin (β0 +Bpart) · cos (C)))

(6.32)

UZ,(X0B,Z0B) = 2rB√12

(max (sin (β0 +Bpart))

−min (sin (β0 +Bpart)))

(6.33)


for the resulting uncertainties at the TCP.

Together with the component and location errors of the linear axes (6.9)-(6.11) the re-

sulting uncertainties at the TCP due to the movement of the X, Y , Z, B and C axis are

evaluated. This leads to

UX = 2√12(EXX2

T CP + EXY 2T CP + EXZ2

T CP + ∆Y 2 · C0Y 2 + ∆Z2 ·B0Z2

+ EXC2T CP + EXB2

T CP + EX2(X0C,Y 0C) + EX2

(X0B,Z0B)

)1/2(6.34)

UY = 2√12(EYX2

T CP + EY Y 2T CP + EY Z2

T CP + ∆Z2 · A0Z2

+ EY C2T CP + EY B2

T CP + EY 2(X0C,Y 0C) + EY 2

(X0B,Z0B)

)1/2(6.35)

UZ = 2√12(EZX2

T CP + EZY 2T CP + EYX2

T CP

+ EZB2T CP + EZ2

(X0B,Z0B)

)1/2(6.36)

6.4.2 Example

For the example of laser cutting figure 6.9 shows a typical 3D-movement including changes

of the orientation of the tool along the TCP path, which leads to large compensation

movements of the linear axes.

Due to the increased number of involved axes, but also caused by the large compensating

movements the uncertainties strongly increase. The deviations in X and Y -direction are

shown in figure 6.10-6.11.

The deviation in Z-direction has no influence on the effective uncertainty in lateral dir-

ection and is not regarded in this example. In figure 6.10-6.11 EXCT CP and EY CT CP are

contributors caused by a positional error ECC. EX(X0C, Y 0C) and EY (X0C, Y 0C) are

contributors due to the location errors X0C and Y 0C of the involved C-axis. Because of

X

Y

X0CY 0C

rC

ϕ0

X

Z

X0BZ0B

rB

β0

Figure 6.8: Description of the location errors of the C and the B axis.


900950

1,0001,050

1,1001,150

700

800

900

1,000500

550

600

650

700

750

TCP Start/End Position

MCS Start/End Position

Position X [mm]Position Y [mm]

PositionZ

[mm

]

rT CP

rMCS

Figure 6.9: 5-Axis movement shown for 5-axis laser cutting.

0 20 40 60 80 100 120 140 160−40

−20

0

20

40

TCP-path [mm]

Deviation

[µm

]

X-direction

EXXEXYEXZC0YB0Z

EXCT CP

EX(X0C,Y 0C)

Ux

Figure 6.10: TCP-deviations in X-direction along the TCP path of figure 6.9


0 20 40 60 80 100 120 140 160−40

−20

0

20

40

TCP-path [mm]

Deviation

[µm

]Y-direction

EYXEY YEY ZA0Z

EY CT CP

EY(X0C,Y 0C)

Uy

Figure 6.11: TCP-deviations in Y-direction along the TCP path of figure 6.9

the large distance between the TCP and the rotation center of these axes the resulting

TCP uncertainties due to these contributors are dominant.

890 900 910 920 930 940 950 960 970 980 990900

910

920

930

940

950

960

970

Position X [mm]

Pos

itio

nY

[mm

]

30

31

32

33

34

35

36

37

Eff

ecti

veunce

rtai

nty

[µm

]

Figure 6.12: Effective TCP-uncertainties for the 5-axis movement from figure 6.9.

108 7. Conclusion and Outlook

Chapter 7

Conclusion and Outlook

This thesis proposes approaches for the solution of two basic problems in trajectory genera-

tion for machine tools. First, a method for geometry optimisation is presented. Secondly, a

method for the exploitation of quasi-redundancy within the given tolerances is introduced.

7.1 Geometry optimisation for machine tools

Chapter 4 showed a quadratic programing approach for the discrete geometry optimisation

(DGO). Within this approach both the geometry and the admissible deviations are discre-

tised in a first step. The objective function minimises the weighted sum of the parametric

derivatives of the different axes while ensuring the geometric constraints and boundaries,

which are given by the manufacturing tolerances. The optimised trajectories are described

by B-Splines and because of the convex problem formulation the optimisation succeeds for

all given optimisation cases.

Contrary to state of the art methods for geometry optimisation the method described above

allows for a point wise limitation of the admissible deviation between the programmed tra-

jectory and the smoothing function. Therefore the tolerance model, which was introduced

in chapter 6 can be incorporated in order to account for the effective geometric uncer-

tainties along the programmed tool path. While state of the art algorithms for geometry

optimisation only minimise geometric parameters like the curvature respectively the curve

rate, the DGO directly minimises parametric derivatives of the assumed B-Spline. These

optimised B-Splines are converted into NC-Code, which is transfered to the NC. The time

discrete feed rate profile is then evaluated by the look ahead inside of the open-loop con-

troller using the optimised parametric derivatives. Based on application examples it was

shown, that this method not only leads to a decrease of the mechanical excitation but also

7.2 Feed rate optimisation using quasi-redundant degrees of freedom 109

to an increase of the feed rate and thus to an increase of the machining productivity.

Regarding the data-flow during machining, which is shown in figure 1.1 the steps from

the design to the set points finally lead to a relative movement between the tool and the

workpiece. These steps are performed subsequently due to the fact that both CAD and

CAM-tools as well as the CNC are independent investigations. State of the art CAM

tools are able to work with NURBS surfaces. These are converted into NC-Code, which in

most cases consists of straight lines. The arising challenge of the smoothing of the given

NC-data, which has to be accomplished by the open-loop controller can be prevented by

a reasonable combination of the CAD and the CAM tool. Approaches in this field are

shown by Lartigue et al. [51] and Farouki et al. [26].

Due to the real time requirement of state of the art open loop controllers the time discrete

feed rate profiles are evaluated subsequently to the geometry optimisation. Usually these

trajectories are not time-optimal in the sense of ”bang-bang”, which means that at each

time-step at least one limitation is at its limit. Therefore further studies should be focused

on a combined optimisation of the feed rate and the geometry, which would further increase

the machining productivity. A next possible step could be the combination of the CAD/-

CAM tool and the set-point generation. This would have the advantage that the complete

data flow during manufacturing mentioned above can be performed and controlled in only

one step. Furthermore the mentioned subsequent steps are based on heuristics.

A combination of all these steps offers the opportunity of an optimisation of the machining

productivity in the sense of a time optimal solution without any heuristics.

7.2 Feed rate optimisation using quasi-redundant de-

grees of freedom

In chapter 5 a method for the exploitation of the quasi-redundancy was introduced. First

of all it was shown, that for a machine tool, which consists of both rotational- and linear

axes within the given orientation tolerance there is no bijective mapping between the

trajectory at the TCP and the trajectory of the different axes. Depending on the regarded

manufacturing process respectively the given manufacturing tolerances this leads to certain

additional degrees of freedom. For a 5-axis laser cutting machine tool it was shown that

there are two additional respectively quasi-redundant degrees of freedom within the given

manufacturing tolerances, which are subject of an optimisation problem.

Assuming that the mechanical excitation of a machine tool is mainly affected by the rate

of force of the different axes, which is proportional to the jerk, the objective function of the

110 7. Conclusion and Outlook

optimisation problem consists of the weighted sum of the jerk of the different axes, which

are involved in the trajectory in response to the inner degrees of freedom. The weights are

chosen proportional to the inertia of the different axes. Dynamic- and geometric constraints

are incorporated by additional constraints and boundaries. With this approach for a given

time discrete trajectory at the TCP jerk minimal trajectories for the different axes are

estimated. Contrary to state of the art trajectory optimisation algorithms in the field of

machine tools within this optimisation approach both the kinematic of the machine tool

and also the inertias of the different axes are taken into account.

Due to the fact that both the gradient and the Hessian of the nonlinear objective function

are given to the solver the optimisation succeeds for most but not for all the given optimi-

sation problems. Because of this and due to the use case of an orientation tolerance of up

to 5◦ a linearisation of the optimisation problem described above is obtained, which leads

to a quadratic programming problem.

For the example of two geometries it was shown that the optimisation described above

leads not only to a decreased jerk of the different axes but also to a rearrangement of

the maximum jerk. In other words: ”The more the inertia of the axis the less the jerk”.

This was achieved by the additional weights described above. By the use of a rigid body

model it was shown that the mechanical excitation of the machine tool and finally path

disturbances could be reduced.

Within the method described above for a given spatial trajectory at the TCP a jerk minimal

trajectory for the machine tool axes is evaluated. To improve this approach further studies

could be focused on a more general approach including a time optimal trajectory at the

TCP and also jerk minimal optimisation of the trajectories of the machine tool axes. This

approach is constrained by both the axes limits and the geometric constraints given by the

manufacturing process. Because all the axes have to be considered simultaneously the size

of the optimisation problem would drastically increase and would take more computational

power.

The method for the exploitation of quasi-redundancy described above is shown for the

example of 5-axis laser cutting. In this case no additional forces induced by the manu-

facturing process need to be regarded during optimisation. Therefore in further studies

the inclusion of these forces should be taken into account, which would lead to a more

general problem formulation, which could be used for the feed rate optimisation in the

field of flank- and end-milling.

The objective function, which is minimised during optimisation is based on the heuris-

tic that a high change of the actuator force respectively the jerk leads to a mechanical

excitation of the machine tool and therefore a dynamic error respectively a decreased ac-

7.2 Feed rate optimisation using quasi-redundant degrees of freedom 111

curacy. Knowing that due to a jerk limitation the productivity decreases further studies

should be focused on the improvement of the limitation during the set point generation.

State of the art methods for set point generation maximise the feed rate respectively min-

imise the machining time considering dynamic limitations like the maximum speed, axes

acceleration as well as the maximum jerk. The dynamic limitations during set-point gen-

eration are given in order to ensure a reliable process and a specified accuracy. Due to

the limited computational power neither a sufficient process-model nor a dynamic machine

model are taken into account during set point generation. The limitations are chosen based

on the heuristics described above. Therefore the investigation of reliable models should be

focused in further studies in order not to suffer from the heuristics described above.

112 A. Parametric study

Appendix A

Parametric study

As mentioned in section 4.3.5 due to the fact that the discrete geometry optimisation is

based on the heuristics of a minimisation of the weighted sum of the parametric derivatives

of a smoothing function r(s) the weighting factors ηi need to be figured out. As already

mentioned in section 4.3.5 due to unintended wiggles during optimisation η3 and η4 for

the weighting of the parametric jerk and the jerk rate are set to zero. The remaining

parameters are η1 and η2, which are figured out in the following by a parametric study.

The coathanger, which is shown in figure A.1 is a very illustrative example, since it contains

both tangent and non tangent transitions between straight lines and circular arcs. Ad-

ditionally the geometry is symmetric and allows for the testing of the symmetric behavior

−20 −15 −10 −5 0 5 10 15 200

5

10

15

20

Position X [mm]

PositionY

[mm

]

Figure A.1: 2D-geometry for the parametric study.

113

of the DGO, which is described in section 4.3.

In the following different tupels for η1 and η2 are given to the DGO in order to evaluate a

smoothing function r(s) within an assumend track tolerance of 50µm. Since the DGO is

based on the heuristics mentioned above r(s) does not necessarily lead to a time optimal

movement. Therefore an alternative criterion for the shape of r(s) is requested. Since the

DGO is primary used for laser cutting, a shape preserving behavior is important. Although

large tolerances for the tool path up to 100µm occur, the contour accuracy is important

for the quality of the manufactured part.

In the following the influence of η1 and η2 on the optimisation result is shown in order

to figure out a tupel satisfying the requirements mentioned above. Since the optimisation

result is influenced by the relationship between ηi,

ξ = η1

η2(A.1)

is introduced. (A.1) with

ξ = {0, 0.001, 0.1, 1, 10, 1000} (A.2)

leads to the optimisation results, which are shown in figure A.2 and are discussed in the

following.

For all ξ ≥ 1 the influence of η1 is dominant. According to (4.60) the arclength L(s) of a

spatial curve is influence by r′(s). Therefore ξ ≥ 1 leads to a minimisation of the arclength

of r(s). As a consequence of that the deviation between r(s) and r(s, ξ = 1)-r(s, ξ = ∞)is not restricted to the immediate vicinity of the transition.

As can be seen from figure A.2 for all ξ ≤ 1 the deviation between r(s) and r(s) is

sufficiently restricted to the immediate vicinity of the tranitions. Since for all ξ ≤ 1 there

is no significant deviation between the obtained solutions for the DGO η1 = 0 respectively

η2 = 1 is assumed.

114 A. Parametric study

−20 −15 −10 −5 0 5 10 15 20

0

5

10

15

20

25

Position X [mm]

PositionY

[mm

]

r(s)r(s, ζ = 0)r(s, ζ = 0.001)r(s, ζ = 1)r(s, ζ = 1000)

Figure A.2: Deviation between r(s) and r(s) due to the parameterisation (A.2).In favor

of the visibility the deviation of the smoothing function in normal direction of the tool path

is scaled two hundred times.

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