Modeling of Flexible Wirings and Contact …...robot linkages and mounting of the wiring in terms of...

Faculty of Civil, Geo and Environmental Engineering

Chair for Computation in Engineering

Prof. Dr. rer. nat. Ernst Rank

Modeling of Flexible Wirings and Contact Interactions in In-

dustrial Robots Using Geometrically Exact Beam Formulation

Reza Barzanooni

Master’s thesis

for the Master of Science program Computational Mechanics

Author: Reza Barzanooni

Supervisor: Prof. Dr.rer.nat. Ernst Rank

Dr.-Ing. Tino Bog

Mohamed Elhaddad, M.Sc.

Date of issue: 15. July 2018

Date of submission: 15. Jan. 2019

Involved Organisations

Chair for Computation in EngineeringFaculty of Civil, Geo and Environmental EngineeringTechnische Universitat MunchenArcisstraße 21D-80333 Munchen

Tebis Technische Informationssysteme AGEinsteinstraße 3982152 , Martinsried/Planegg

Declaration

With this statement I declare, that I have independently completed this Master’s thesis. Thethoughts taken directly or indirectly from external sources are properly marked as such. Thisthesis was not previously submitted to another academic institution and has also not yetbeen published.

Munchen, January 15, 2019

Reza Barzanooni

Reza Barzanooni

Abstract

Due to their speed and precision, industrial robots are used in a wide range of tasks in modernfactories. Already, there exist simulation tools to plan kinematically admissible movementsfor this kind of robots. However, such tools do not take deformation of the wiring mountedon robots’ links into consideration, although damage of the wiring during robot motion caninterrupt the whole process. In this thesis, the elastic behavior of the wiring and its contactinteractions with the robot links are modeled to define critical stress and deformation statesduring the robot movement.Nonlinear Finite Element Analysis is used to compute deformation of the wiring. The wiringis modeled as a beam structure which undergoes large deformations during the robot mo-tion. However, its inertia effects are neglected. Also, due to its low slenderness ratio, sheardeformations are not considered. A geometrically Exact beam element of Kirchhoff type isutilized to model the nonlinear beams. This type of element is based on the nonlinear shear-deformable Simo-Reissner beam theory, and shear deformations are excluded by enforcingthe Kirchhoff constraints. Additionally, due to low torsional stiffness of cables, a simplifiedversion of the element which excludes torsional deformations is applied.Furthermore, in the contact model, since friction effects between the wiring and the robotarms are negligible, it is not considered. Also, the robot links are assumed as analytic surfaces.The non-penetrating contact condition is enforced using the penalty method and discretizedby the Gauss Point to Analytic Surface approach.To solve the resulting nonlinear system of equations, the Newton-Raphson method with thedisplacement control path-following technique is employed. Numerical examples showcasethe robustness and computational efficiency of the proposed method.

Acknowledgment

During my bachelor studies, I had the opportunity to do an internship in a group who wasdesigning a 6R industrial robot. At the final stages of the design process, a question arose thattook much longer time to answer than we first thought. The question was how the wiringshould be mounted on the robot arms so that it does not get damaged during the robotmotion. Because there was no well-known engineering tool to assist us to deal with thisproblem, we had to go through a long try and error process and use our “engineering senses”to come up with a good answer. Since then, I was curious about a systematic approach todeal with this problem. After some time, in my master studies, when we had to choose atopic for a course project named “software lab”, one of available options was to work on acode to deal with the exact same problem using computational mechanics approach. As aresult, I jumped for the topic, and had the chance to work on it with two hardworking andsmart teammates. The project was successful as a proof of concept. Hence, I was interestedto continue working on this topic as my master research project in order to further improvethe idea.I would like to express my gratitude to all the amazing teammates with whom I had theopportunity to work during this journey. For sure, I have learned a lot from them, and thiswork was not possible without them. I am thankful to Prof. Ernst Rank who acceptedto supervise and examine this work at the chair of “Computation in Engineering”. Also, Iwould like to thank “Tebis AG” for their partnership in both projects, the software lab andthe master thesis, specially the colleagues at the “Manufacturing Technology” departmentand head of the department Dr. Alexander Muthler, for providing an inspirational andsupportive working environment. My special thanks go to my two supervisors Dr. Tino Bogand Mohamed Elhaddad for their support and patience during the both projects. All theparts of their supervision were interesting for me, but if I had to pick one, I would refer tothe discussions at our meetings and their aspiration to help me to understand all aspects ofthe topic.I would like to take this opportunity to thank my family for their unconditional love. Mystrongest hope in life is to know that I have two wonderful parents who have supported mein all steps of my life. You two are the best parents I could ever ask for.Finally, I would like to ask the dear readers not to hesitate to contact me if they have anypoints to mention regarding the content of this work. Although this project has been finished,my learning journey continues. I would appreciate such points and feedbacks.

IX

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Continuum Mechanics 5

2.1 Nonlinear Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Kinematics of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Work and Energy Theorems . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.4 Material Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Nonlinear Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Weak Form of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.4 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Nonlinear Solver and Path Following Techniques . . . . . . . . . . . . . . . . 15

3 Review of Nonlinear Beam Elements and Contact Formulations 21

3.1 Nonlinear Beam Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Contact Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Contact Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.2 Contact Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.3 Beam Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Geometrically Exact Beam 29

4.1 Geometrically Exact Simo-Reissner Beam Theory . . . . . . . . . . . . . . . . 29

4.2 Geometrically Exact Kirchhoff Beam Theory . . . . . . . . . . . . . . . . . . 31

4.2.1 Torsion Deformable GE Kirchhoff Beam . . . . . . . . . . . . . . . . . 32

4.2.2 Torsion Free GE Kirchhoff Beam . . . . . . . . . . . . . . . . . . . . . 35

4.3 Geometrically Exact Kirchhoff Beam Element . . . . . . . . . . . . . . . . . . 36

4.3.1 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . 36

4.3.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.3 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.1 Example 1: Pure bending in 2D . . . . . . . . . . . . . . . . . . . . . 41

4.4.2 Example 2: 3D line loading . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Beam to Surface Contact 45

5.1 Closest Point Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Penalty Method vs. Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . 475.3 Contact Enforcement Using Penalty Method . . . . . . . . . . . . . . . . . . . 47

5.3.1 Penalty Potential and weak form . . . . . . . . . . . . . . . . . . . . . 475.3.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . 485.3.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3.4 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.4.1 Example 1: Beam Patch Test . . . . . . . . . . . . . . . . . . . . . . . 505.4.2 Example 2: Twisting of a Beam Against a Rigid Cylinder . . . . . . . 51

6 Modeling Interactions of the Wiring and Robot Linkages 55

6.1 Robot Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.1.1 Denavit-Hartenberg Notation . . . . . . . . . . . . . . . . . . . . . . . 566.1.2 Robot Transformation Matrices . . . . . . . . . . . . . . . . . . . . . . 57

6.2 Robot and Wire Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.2.1 Calculation of Cable Boundary Conditions . . . . . . . . . . . . . . . . 616.2.2 Adaptations to Displacement Control Solver . . . . . . . . . . . . . . 626.2.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7 Summary and Outlook 69

A Closest Point Projection to Analytic Surfaces 73

A.0.1 Projection onto Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A.0.2 Projection onto Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A.0.3 Projection onto Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 74A.0.4 Projection onto Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B Analytic Solution for Beam Contact with a Rigid Cylinder 77

1

Chapter 1

Introduction

1.1 Motivation

Industrial robotic arms play a crucial role in automation of production lines. Due to theirability to perform rather complex movements with high speed and precision, they are widelyused in modern factories. Typical applications of such robots are welding, painting, assem-bling, production inspection, and testing. Already, there exist simulation tools that helpengineers to plan kinematically admissible trajectories for industrial robots. However, thesetools do not take deformation of the wiring mounted on their links into consideration, al-though there are numerous movement configurations that can damage the wiring and henceinterrupt robots’ tasks or even the whole production line. The development of a tool thatincorporates robot wiring deformation into the kinematic simulation can assist engineers withbetter motion planning, or in a higher level, with coming up with new optimized designs ofrobot linkages and mounting of the wiring in terms of minimal limitation of robot maneuver-ability by the wiring.

In the current work, kinematic modeling of robot movement is extended with the simulationof wiring elastic behavior in order to define critical stress and deformation states during oper-ation. The following requirements are essential for the modeling of the wiring deformations:

• In virtual reality applications, spring-mass models are typically used for flexible struc-tures. Although such models are based on physics, here in order to have a correctrepresentation of stress state, our model should be based on more realistic physicaltheories.

• Industrial robots are able to perform complex maneuvers which can cause complicatedstates of deformation in the wiring— mostly geometrically nonlinear. The model shouldtake into account such non-linearities.

• Unlike existing models for ropes and chains, bending effects play an important role inthe deformation of cables. Hence, it should be considered.

• Another crucial factor in deformation of the wiring is the contact interactions withrobot arms. However, friction between them is negligible.

2 1. Introduction

• Our simulation tool should be able to model typical boundary conditions that happenin real world attachment of wiring to robot linkages.

Additionally, since the developed model is going to accompany kinematic simulation tools,which are mostly fast and robust, it is desired that the cable simulator causes minimumpossible overhead in terms of solution speed and robustness.

We model wires as beams which undergo large deformations. Although at different time pointsduring the motion, the state of the deformation is modeled, inertia effects are not considered(quasi-static simulation). Also, due to low slenderness of the wiring, shear deformationsare negligible. In the first part of this work, large strain beam finite element method (FEM)elements that can robustly model the geometrically nonlinear behavior of the wiring is studied.Additionally, to efficiently model the contact interactions between wiring and robot arms,beam contact methods are subject to study in this section. In contact algorithm, robotlinkages are considered as analytic surfaces and as mentioned in the 4th requirement above,friction between two contact bodies, the wiring and robot links, are neglected. In the finalpart, robot kinematics is investigated and extended with cable simulator.

1.2 Outline

The following report is structured into three main parts: the first part consists of the funda-mentals of FEM, beam elements and contact mechanics. In the second part, the two mainsub-modules of the cable simulator are presented. Finally, in the last part, it is integrated tothe simulation of robot motion.

In chapter 2 the general fundamentals of this work are laid. After a short review of nonlinearcontinuum mechanics and the most important theories, displacement based nonlinear FiniteElement Analysis is discussed. Application of this method to differential equations leads toa system of nonlinear equations that should be solved by iterative solvers, which is treatedin the final section of the chapter.

Chapter 3 treats beams as 1D structural elements. First, existing beam theories and finiteelement beam elements are reviewed. Also, membrane locking, a locking effect that happens inshear-free beam elements, is introduced here. Afterwards, fundamental knowledge of contactmechanics including contact enforcement techniques and contact discretization methods ispresented. A review of research works on beam contact is also presented.

Chapter 4 is devoted to beam element formulation. Simo-Reissnear beam theory (also knownas Geometrically Exact theory) and its kinematics are first briefly presented. However, thistheory accounts for shear-deformations as well. Next, Kirchhoff constraints are applied to thetheory which results in geometrically exact shear-free beam theory and is further simplified toexclude torsion deformations. The simplified version is suitable for modeling the deformationsin cable-like structures. In the subsequent part, derivation of FEM beam element formulationbased on the torsion-free theory and cubic Hermitian polynomials is presented. Finally, twonumerical studies are performed to investigate accuracy and convergence behavior of thebeam element.

In chapter 5 beam to analytic surface contact is discussed. Penalty regularization method isused to derive contact potential. A Gauss-point to analytic surface is used for discretization

1.2. Outline 3

of the potential. After complete derivation of the contact element using nonlinear FEMmethod, two studies are performed to examine its numerical behavior.

In the final chapter, the developed cable simulator is used to model deformation of thewiring while robot arms move. To this end, boundary conditions for the wiring shouldbe calculated from robot kinematics, for which the so-called DH notation is used. Specialpractical difficulties of modeling the interactions are also discussed in this chapter.

4 1. Introduction

5

Chapter 2

Continuum Mechanics

This chapter lays the theoretical foundation of subsequent sections. Problems studied in com-putational mechanics are described by a differential equation that is derived from continuummechanics. In most cases, there are no general closed form solutions for this equation, andtherefore should be solved using numerical methods. Along other methods, the Finite Ele-ment Method (FEM) is well-established, particularly in computational solid mechanics. Inthis method, the problem domain is divided to smaller elements and the solution is obtainedat certain points along these elements (element nodes). Applying the finite element methodto a differential equation results in a system of equations that should further be solved.

Phenomena that involve nonlinearities are described and solved by use of nonlinear continuummechanics theories and nonlinear FEM. In solid mechanics, nonlinearities are categorizedinto geometric, material, or load nonlinearity. In this thesis, only the first one is considered.Another major difference between linear and nonlinear problems is that the resulting systemof equations should be solved by iterative solution techniques.

In the following, after a brief introduction to the nonlinear continuum mechanics, FEMprocedures for numerical computation are discussed. Finally, the Newton-Raphson methodas an iterative solution technique in general, as well as different path-following methods inparticular are discussed.

2.1 Nonlinear Continuum Mechanics

Continuum mechanics deals with kinematics of motion and mechanical behavior of mediaassuming them as continuum mass rather than discrete particles. The behavior of such mediais described by governing differential equations which are derived from kinematic assumptionsand conservation laws, namely conservation of mass, momentum and energy. Informationabout media may be added to the model by constitutive laws. Here, the basics of nonlinearcontinuum mechanics are treated. More extended information can be found in [BW08] .

6 2. Continuum Mechanics

x

y

z

dX

Q

P

dx

q

pxqXP

XQ

xp

β0

βt

Figure 2.1: General motion of arbitrary body β and relevant variables

2.1.1 Kinematics of Motion

Kinematics provides media’s geometrical description of motion. To this end, a body β at itsinitial configuration (time t= 0, β0) and current configuration (time = t, βt) is considered,(Fig. 2.1). Particles coordinates in the body at initial and current configurations are denotedbyX and x respectively. Motion of the body can be described in general form by the functionφ defined as,

x = φ(X, t). (2.1)

The mapping between the undeformed and deformed configurations can be obtained by con-sidering a constant value of t in (2.1), whereas a constant value of X represents trajectory ofa particular particle on the body.

The displacement of body particles u is defined as,

u = x−X. (2.2)

In infinitesimal deformation analysis, the displacement u is limited to values much smallerthan the body’s dimensions, so that geometrical changes can be neglected. However, this isnot the case in finite deformation analysis, and geometrical nonlinearities are therefore theresult.

In the finite deformation analysis, a careful distinction between the so-called material andspatial configurations is essential. The former describes the position of the body before de-formation and the latter describes where the body is after deformation. In literature, theseconcepts are also referred to as Lagrangian and Eulerian descriptions, respectively. In otherwords, the material description concerns with behavior of a material particle, whereas thespatial description relates to behavior at a spatial position. The governing equations dealingwith the current configuration, as a result, should be formulated using spatial description.

2.1. Nonlinear Continuum Mechanics 7

Nevertheless, the material description is inevitable to consider constitutive behavior of ma-terial in solid mechanics problems.

In the finite deformation analysis, the deformation Gradient tensor F is used to relate quan-tities in initial and current configurations. This tensor allows to describe the relative positionof two material particles after deformation with respect to their relative position before de-formation. As a result, it is essential for deformation and strains definition.

Consider material particles Q and P at time t = 0 (Fig. 2.1) with their relative position dX,

dX = XQ −XP . (2.3)

The two particles’ corresponding deformed configurations are q and p at time = t. Using(2.1) the relation between the deformed and undeformed positions is established as,

xq = φ(XQ, t), xp = φ(XP , t). (2.4)

Consequently, the deformed relative position vector dx can be obtained as,

dx = xq − xp = φ(XQ, t)− φ(XP , t). (2.5)

The above-mentioned Deformation Gradient tensor F defines how the relative position vectorchanges after deformation, hence

F =dx

dX= ∇φ. (2.6)

In research literature, the relation between quantities in spatial and material descriptions, e.g.dx and dX respectively, is expressed by the so-called push-forward and pull-back operators.In this manner, dx is the push-forward equivalent of dX and is computed from deformationgradient tensor definition (2.6) as,

dx = F dX (2.7)

or vice versa, dX is pull-back to the material description of dx, which can be defined by theinverse of deformation gradient F−1,

dX = F−1dx. (2.8)

2.1.2 Equilibrium Equations

In this section, we confine our attention to the derivation of static equilibrium of forces in adeformable body in spatial description. The body volume and boundary area are defined byυ and ∂υ. Furthermore, it is under action of body force f and traction force t. Using Cauchystress measure, force per unit current area, the traction force is expressed as t = σn, wherethe vector n is normal to the surface. The equilibrium of linear momentum implies that thesum of forces on the body should vanish. Hence,

∫

∂υ

σnda+

∫

υ

fdv = 0. (2.9)


Here, the first term, which is the integral of traction forces over the boundary, can be trans-ferred to a volume integral using Gauss theorem. This results in,

∫

υ

(divσ + f)dυ = 0 (2.10)

where, divσ denotes divergence of the second order Cauchy stress tensor,

divσ = ∇σ : I =3∑

i,j=1

δσijδxj

ei (2.11)

Expression (2.10) should hold on each arbitrary region of the body:

divσ + f = 0. (2.12)

The equilibrium equation given above should hold for each point on the body. Due to thisfact, it is referred to as strong form of equilibrium as opposed to the weak form of equilibriumthat is treated in Sec. 2.1.3.

In addition to equilibrium of linear momentum, or forces, that is discussed so far, equilibriumof angular momentum should also be satisfied within the body. Based on which, total momentof body and traction forces around each arbitrary point should vanish. When formulated,this results in the symmetry of Cauchy stress tensor as shown in [DBCRV12], [BW08].

2.1.3 Work and Energy Theorems

In the last section, the strong form of equilibrium is derived. As mentioned, the strong formholds point-wise within the deformable body, as opposed to the weak form which only shouldbe fulfilled in the integral sense on the domain. Two convenient tools to derive weak form ofequilibrium are the principle of virtual work, and the principle of minimum total potentialenergy.

Principle of Virtual Work

To derive the virtual work equation, we consider vector of residual forces as sum of externaland internal forces applied to a sub-region of the body. Based on the equilibrium of forces,the residual should be zero.

r = divσ + f = 0 (2.13)

Furthermore, consider δv to be a virtual velocity. It can have any arbitrary value and onlyhas to vanish at Dirichlet boundary. The virtual work per unit volume and time done byresidual force (2.13) is,

δw = rδv. (2.14)

2.1. Nonlinear Continuum Mechanics 9

Since δv is arbitrary, strong form of equilibrium (2.12) implies virtual work (2.14) should bezero. Integration over deformed body volume v results,

δW =

∫

v

rδvdv =

∫

v

(divσ + f)δvdv = 0. (2.15)

Substituting the traction force t = σn and using divergence and Gauss theorems as shownin [BW08], leads to the final form of spatial virtual work equation as,

∫

v

σ : ∇δv −

∫

v

f · δvdv −

∫

δv

t · dvda = 0. (2.16)

Here, the first term expresses virtual work of internal forces, while the second and thirdterms are contributions of external forces. Consequently, the equation of virtual work can bewritten in general form as,

δW = δW int − δW ext. (2.17)

Furthermore, taking into account the symmetry of the Cauchy stress tensor, the gradientof virtual velocity, ∇δv, in the internal virtual work can be substituted by virtual rate ofdeformation, δd,

δW int =

∫

v

σ : δd. (2.18)

For detailed derivation, interested reader might refer to [BW08].

Remark 1. Work Conjugacy: In (2.18), contraction of σ and virtual d results in virtual workof internal forces. Cauchy stress tensor, σ, and rate of deformation tensor, d, are thereforecalled work conjugate pairs. In the weak form, using the right work conjugate strain measurewith the selected stress tensor is vital in order to calculate the correct value of internal forces.

Principle of Minimal Total Potential Energy

Elastic energy as a function of deformation measure C stored per unit volume in a body isΨ(C) considering a hyperelastic material. This concept is further discussed in Sec. 2.1.4.Also, we assume external forces applied to the body are conservative and path-independent.The total potential energy at deformation state φ is,

π(φ) =

∫

V

Ψ(C)dv −

∫

V

f · φdV −

∫

δV

t · φdA. (2.19)

Based on the principle of minimum total potential energy, the deformation state φ thatminimizes functional (2.19) is the equilibrium state. The variation of π, δπ, can be used tocalculate its minimum.

In [BW08] and [Wri08], it is proved that the variation of the total potential energy and theweak form resulting from the principle of virtual work are equivalent,

δπ = δW . (2.20)


The formulation of the minimal total potential energy is beneficial since it provides an addi-tional tool to derive week form of equilibrium. Additionally, topics such as incompressibility,contact boundary conditions, and anti-locking mechanisms can be treated using this ap-proach. In this work, the contact contribution to the weak formulation of the problem isformulated in terms of contact potential energy in Chapter 5.

2.1.4 Material Laws

The equilibrium equations derived in the last section express internal forces in body in termsof stress. These are secondary variables which in turn result from body deformations. Hence,material laws are essential to describe the relation between stresses and measures of bodydeformations, strains.

Hyperelasticity is a widely used class of material equations and base for more complex ma-terial laws such as elastoplasticity, viscoelasticity, and viscoplasticity. A material is termedhyperelastic, if stresses are path-independent, or in other words only depend on initial andcurrent positions. In this case, stresses can be derived from an elastic energy potential, Ψ.

Usually, the energy potential Ψ(C(X),X) is defined as a function of the right Cauchy-Greenstrain tensor C in the material configuration, where,

C = F TF . (2.21)

Here, F denotes the deformation gradient. It is to be noted that Ψ should be objective, andhence must not change when the body goes under action of rigid body deformations.

Taking into account the path-independence of stresses, the energy potential Ψ is equal tothe work done by work conjugate stress-strain pair from the initial position to the currentposition (e.g. second Piola-Kirchhof stress tensor, S, and Green-Lagrange strain tensor, E).Consequently, the second Piola-Kirchhoff stress tensor is derivative of Ψ with respect to theGreen-Lagrange strain tensor,

S =∂Ψ

∂E(2.22)

The relation between S and E is generally nonlinear. However, as shown in [BW08], lin-earization can lead to a linear relation between directional derivative of S and E. KnowingE = 1/2(C − I), the material elasticity tensor, C, can be established as,

C =∂S

∂E= 2

∂S

∂C=

4∂2Ψ

∂C∂C(2.23)

where C is a fourth order tensor.

St. Venant-Kirchhoff material law

St. Venant-Kirchhoff materials are a special type of hyerelasticity with a simple potentialenergy function. These are especially suitable for small strain cases.

2.2. Nonlinear Finite Element Analysis 11

Only two material parameters are necessary, namely the Lame parameters λ and µ, to definethe potential energy function of this type of hyperelasticity,

Ψ(E) =1

2λ(trE)2 + µE : E (2.24)

Based on (2.22) and (2.23) second Piola-Kirchhoff stress and coefficients of elasticity tensorcan respectively be defined as,

S = λ(trE)I + 2µE (2.25)

CIJKL = λδIJδKL + 2µδIKδJL (2.26)

here, I denotes second order identity tensor and Kronecker delta is shown by δij .

The material elasticity tensor denoted by (2.26) represents the classical small strain linearelasticity theory. The relation between the Lame parameters and the well-known materialparameters, Young’s modulus E and Poisson ration µ, are defined as,

λ =νE

(1 + ν)(1− 2ν)(2.27a)

µ =E

2(1 + ν). (2.27b)

2.2 Nonlinear Finite Element Analysis

In computational solid mechanics, the Finite Element Analysis (FEA) is an establishedmethod to solve differential equations. The pure displacement version of FEA, where pri-mary variables are displacements, is simple and straightforward in majority of linear andnonlinear solid mechanics applications. Hence, here we lay the foundation for displacement-based nonlinear finite element analysis. General procedure of an FEA analysis and specialconsiderations one must take into account in finite element formulations are discussed.

2.2.1 Weak Form of Equilibrium

Finite element formulations are based on weak form of differential equation at hand. In Sec.2.1.3, principles of the virtual work and the minimum total potential energy were presentedas tools to derive the weak form of equilibrium equations.

In nonlinear problems, the weak form of equilibrium can be nonlinear with respect to eithergeometry or material, or both. Hence, an iterative approach should be utilized to obtainequilibrium. This makes the linearization of the weak form vital. There are two commonapproaches in derivation of finite element formulations, first linearization then discretization,or vice versa. Here, the former is adopted as is more common for solid continua.


2.2.2 Linearization

The weak form of equilibrium is linearized using the concept of directional derivatives.

Directional Derivative of Nonlinear Equation

Consider a set of arbitrary nonlinear equations

F (x) = 0, (2.28)

where x is a set of variables. The directional derivative of F at fixed value of x = xo in generaldirection u, DF (x0)[u], defines how the function F changes when there is an increment inx0 in the direction u. To do so, a function f with a single parameter ǫ is defined as,

f(ǫ) = F (x0 + ǫu). (2.29)

The change of F is equivalent to the limit of the change of f as ǫ → 0, which can beapproximated by use of Taylor’s series expansion,

f(ǫ) = f(0) +df

dǫ

∣∣∣∣ǫ=0

ǫ+1

2

d2f

dǫ2

∣∣∣∣ǫ=0

ǫ2 + · · · . (2.30)

Substituting (2.29) into the above equation and neglecting nonlinear terms results in,

F (x0 + ǫu)− F (x0) ≈ ǫdF (x0 + ǫu)

dǫ

∣∣∣∣ǫ=0

(2.31)

where the right hand side is a linear approximation of the change in F . It should be notedthat parameter ǫ is introduced as an auxiliary variable to perform derivations. Hence, toeliminate it from the above we can put ǫ = 1 (see e.g. [BW08]),

DF (x0)[u] =d

dǫ

∣∣∣∣ǫ=0

F (x0 + ǫu) (2.32)

Approximation (2.32) is linear with respect to u.

Linearization of the Weak Form

The weak form derived in Sec. 2.1.3 is a function of the equilibrium state φ and the virtualvelocity vector δv. Furthermore, as mentioned earlier, the virtual work can be split in twoparts, namely the virtual work of external forces and the one from internal forces. Hence,

δW (φ, δv) = δW int − δW ext = 0. (2.33)

2.2. Nonlinear Finite Element Analysis 13

Consider a trial equilibrium state φk. Linearization of the weak form above at trial equilibriumstate φk can be denoted in general form as,

δW (φk, δv) +DδW (φk, δv)[u] = 0. (2.34)

Furthermore, plugging (2.33) into the linearized term of (2.34) results in:

DδW (φk, δv)[u] = DδWint(φk, δv)[u] −DδWext(φk, δv)[u] (2.35)

2.2.3 Discretization

Discretization of the linearized weak formulation results in a system of linear equations. Inthis case, the solution is obtained on a limited number of discrete points, namely the nodesof the finite element mesh.

The isoparametric concept refers to using the same shape functions for approximation ofgeometry and displacement field. Interpolation of the initial geometry, X, in terms of initialposition of element nodes, Xi, is,

Xh =n∑

i=1

NiXi, (2.36)

where, n is number of the nodes and Ni represents the shape functions. In general, dis-cretization of solids in 3D space requires three variable parameter space, hence Ni(ξ1, ξ2, ξ3),whereas for descretization of beam centerlines only one parameter, ξ, would suffice. Thesubscript −h in this work denotes discretized variables.

Referring to the isoparametric approach, the discretization of displacement filed u reads,

uh =

n∑

i=1

Nidi. (2.37)

Here, the nodal displacements are denoted by d. Taking into account approximations (2.36),(2.37) and relation (2.2), the current particle positions x can also be discretized as,

xh =

n∑

i=1

Nixi. (2.38)

Differentiating the above equation with respect to time, interpolation of the velocity and itsvariation are obtained as,

vh =

n∑

i=1

Nivi, δvh =

n∑

i=1

Niδvi. (2.39)


The deformation gradient F as shown in (2.6) results by differentiating (2.38) with respectto the initial geometry,

Fh =

N∑

i=1

xi ⊗∇0Ni, (2.40)

where, ∇0Ni is differentiation of shape functions with respect to the initial geometry, ∇0Ni =∂Ni/∂X. It can be related to ∇ξNi = ∂Ni/∂ξ via inverse of the Jacobian matrix J ,

∇0Ni =∂Ni

∂X=

(∂X

∂ξ

)−1 ∂Ni

∂ξ= J−1∇ξNi. (2.41)

In component form, J can be expressed as,

J =∂X

∂ξ=

∂X∂ξ1

∂X∂ξ2

∂X∂ξ3

∂Y∂ξ1

∂Y∂ξ2

∂Y∂ξ3

∂Z∂ξ1

∂Z∂ξ2

∂Z∂ξ3

(2.42)

Other strain measures interpolations can be obtained either from (2.40) or the same manneras deformation measure F discretized.

Inserting the derived forms of discretization at arbitrary node i into the weak from (2.33)results in,

δWh(φk, Niδvi) = δviri, (2.43)

where ri is denoted as vector of residual forces at node i. In a similar manner, discretizationof the linearization of the weak form (2.34) at element j that connects nodes i and i+ i canbe performed as,

DδWh(φk, Niδvi)[Ni+1di+1] = δvi ·Kji,i+1dj . (2.44)

The matrix Kji,i+1 in the above discretization is termed as tangent matrix of element j.

The isoparametric shape functions Ni should satisfy a series of conditions that are resulteddirectly form interpolation property or convergence requirements of numerical solutions.[Fel04]. Next, these requirements are briefly discussed.

Requirements of Shape Functions

For mathematical statement of some conditions treated in this interlude, first, the concept ofthe variational index should be defined. The highest spatial derivative of displacement thatappears in the weak form of equilibrium is called the variation index and denoted by m inthis section.

Requirements that shape functions Ni must satisfy are as follows:

1. interpolation condition: the shape function Ni must take a unit value at node i andzero value at other nodes.

2. partition of unity: Sum of all shape functions at each point equals to one.

2.3. Nonlinear Solver and Path Following Techniques 15

3. completeness: In numerical methods, convergence property requires that at the limitof a mesh refinement process the numerical solution captures the analytic one. To thisend, the shape functions should be m−complete, they must be able to represent all thepolynomial terms of order ≤ m.

4. compatibility: As the elements deform in an FEM simulation, we should make surethat no gaps appear in boundary nodes. In other words, displacement field shouldbe continuous. Accordingly, shape functions must be Cm−1 continuous at elementboundary and Cm inside each element.

The first two requirement are resulted from interpolation properties while the other two followfrom convergence requirements. [Fel04]

2.2.4 Numerical Integration

The final step in FEA is the evaluation of integrals, e.g. tangent matrix (2.44) or residualforces vector (2.43). Since at most cases exact integration is not possible or complicated todeal with, the integrals are treated numerically at element level using Gauss quadrature rule.Numerical approximation of function f(ξ), ξ ∈ [−1, 1] reads,

∫ 1

−1f(ξ)d(ξ) =

nGP∑

i=1

ωξif(ξi) (2.45)

here, ξi and ωξi denote Gauss points and weights respectively. nGP is number of Gauss points.If f(ξ) is a polynomial function of order p, the number of Gauss points required for exactintegration is

nGP =p+ 1

2. (2.46)

It should be noted that the numerical integration (2.45) is performed on integration spaceξ ∈ [−1, 1]. For integrand f(x) with a different parameter space, transformation to theintegration space results,

∫

(x)f(x)dx =

∫ +1

−1f(ξ) det J(ξ)dξ, J(ξ) =

dx

dξ. (2.47)

The right hand side of the above expression can further be evaluated by the Gauss rule (2.45).

2.3 Nonlinear Solver and Path Following Techniques

In continuum mechanics, nonlinear phenomena are solved by linearizing the governing equa-tion and then solve the resulting linear equations in an iterative manner. The Newton-Raphson method is a well-established method for such iterative solution techniques. In thefollowing, this method is briefly introduced.

To establish Newton-Raphson method consider the set of nonlinear equations F (x) intro-duced in (2.28) with an initial guess x0. Furthermore, we consider a general change increment


in the solution u that leads to x = x0+u closer to the real solution of (2.28). Approximatingthe change in F with directional derivative relation in (2.32), F (x0 + u) is,

F (x0 + u) = F (x0) +DF (X0)[u]. (2.48)

Additionally, referring to (2.28) the above relation results in,

F (x0) +DF (x0)[u] = 0. (2.49)

Solving the above system of linear equations for u and increasing the resulted solution xk

with u iteratively, the Newton-Raphson solution procedure can be written as,

DF (xk)[u] = −F (xk), xk+1 = xk + u, (2.50)

where k denotes the iteration number. This procedure can be repeated until a convergencecriteria is met. Usually, the Euclidean norm of the solution increment ‖uk+1‖ or the residualvector ‖Gk+1‖ are checked for convergence. The solution is converged if these norms fallbelow pre-defined tolerances δR, δu, i.e,

‖uk+1‖ < δu, ‖Gk+1‖ < δR. (2.51)

In the procedure described above, the concept of load stepping is not considered. Althoughin some cases it might be possible to achieve a direct solution for a given external load, instandard nonlinear FEA the load is applied via a series of increments (load steps). This inturn causes easier convergence to the final solution. In case of using hyperelastic materialconstitutive model, final stresses do not depend on how the load is applied. However, thereare other material models which are path-dependent. These two facts motivate the conceptof path-following methods.

To treat the concept of path-following techniques, we reformulate the nonlinear system ofequations in (2.28) as,

G(u, λ) = R(u)− λP (2.52)

where G, R, and P denote residual, internal and external forces respectively. In order toenable us to apply the external load P in a series of increments a load factor λ is employed.

The load-displacement curve in Fig.2.2.a shows an arbitrary solution path to (2.52). Themost common path-following techniques in the nonlinear FEA are load control, displacementcontrol, and different variants of arc-length method. Each of these methods add a constraintequation, f(u, λ) = 0, to (2.52) which defines the load factor at each load step.

In this work, in context of nonlinear solver superscript i and subscript k of a variable denoteload step and iteration number respectively. Hence, e.g. uik is read displacement at load stepi iteration k.


λ

u

(a)λ

u

(b)

λ

u

(c)λ

u

(d)

λi

λi+1

ui ui+1 ui ui+1

∆s

Figure 2.2: Path following methods [Wri08], (a) Load-displacement curve for an arbitrary solutionpath, (b) Load control Method, (c) Displacement control method, (d) Arc-Length method

Load Control Method

The constrain equation for this method is described as,

f = λi − λi, (2.53)

where, λ denotes prescribed value of λ. As a result, at each solution step, the load factor isconstant. The constraint equation can be shown in the load displacement curve by a parallelline to the displacement axis with λ = λ. The nonlinear set of equations (2.52) at load stepi can be written as,

G(u) = R(u)− P ∗, P ∗ = λiP , (2.54)

which is the standard Newton-Raphson solution technique. The load control method is illus-trated in Fig.2.2.b. This is an easy to implement and practical method. However, solutionsbehind a local extremum cannot be reached by this method.

Displacement Control Method

Unlike load control, in this method displacement increments at each solution step are pre-scribed. Thus, the constraint equation is,

f = ui − ui. (2.55)


In [DBCRV12], authors argue that the displacement control is preferred to the load controldue to the following facts. First, the tangent stiffness matrix resulting from this method isbetter conditioned than the one in load control method. Additionally, load control tangentstiffness matrix K becomes singular at limit points and local extrema, whereas this is not thecase for the displacement method. Better convergence properties of the displacement methodcan be geometrically proved from Fig. 2.2. For each displacement state u∗ a correspondingstate on the solution path can be found, while for load factors higher than the limit loadthere is no state.

Although Fig. 2.2.c illustrates the displacement control algorithm with one DOF, it is possibleto apply this method to cases where a group of non-zero displacements are prescribed in anincremental manner as shown in [DBCRV12]. To this end, we assume discretized system oflinear equations to be solved is,

K∆d = fext − fint (2.56)

herein d is vector of nodal displacements and similar to the last section that tangent stiffnessmatrix and force vector are denoted byK and f respectively. Incremental displacement vector∆d can be decomposed to prescribed DOFs, ∆ap and the ones that are not prescribed, orfree DOFs, ∆af and should be calculated,

∆d =

[∆df

∆dp

]

. (2.57)

Based on the above decomposition, the system of equations (2.56) can be partitioned as,

[Kff Kfp

Kpf Kpp

] [∆df

∆dp

]

=

[(ff )ext

0

]

−

[(ff )

0int

(fp)0int

]

(2.58)

It is worth noting that at the prescribed DOFs no external forces should be applied. Thisremark is considered at (2.58) where external forces of ∆dp is set to zero. To calculateunknown free DOFs, ∆df , at first iteration the prescribed DOFs can be eliminated from theabove system of equations, which yields to,

∆d1f = K−1

ff

((ff )ext − (ff )

0int −Kfp∆dp

), (2.59)

where Kfp∆dp can be noted as force vector applied to free DOFs due to the displacementof prescribed ones. For next iterations, ∆ap vanishes and therefore expression (2.59) can besimplified to,

∆dj+1f = K−1

ff

(

(ff )ext − (ff )jint

)

. (2.60)

The procedure treated so far involves rearrangement of the stiffness matrix and eliminationof some entries. Hence, this method is termed as a reduction technique. Reduction canbe computationally expensive for large matrices. Rearranging can be avoided by two ap-proaches. First, in the assembly process only two sub matrices Kff and Kfp are generated.Second, modifying the global stiffness matrix instead of rearranging it as shown in [Fel04].Modification can be beneficial when a few DOFs are prescribed.


The idea behind this approach is to directly modify entries of the K matrix, so they resultin zeros for ∆dp and same solution for ∆df as (2.59) and (2.60). This can be realizedby substituting off-diagonal entries on rows and columns corresponding to the prescribedDOFs with zeros, and diagonal entries with ones. Then, equivalent external forces due to∆dp for free DOFs should be added to the right hand side, and the right hand side entriescorresponding to ∆dp are set to zero. For clarity the process is explained on partitionedsystem of equations (2.58), although partitioning is not explicitly necessary. At the firstiteration, free DOFs are obtained as,

[∆d1

f

0

]

=

[Kff 0

0 I

]−1([(ff )ext

0

]

−

[(ff )

0int

0

]

−

[Kfp∆dp

0

])

. (2.61)

For the next iterations similar to (2.60) ∆dp vanishes and therefore

[

∆dj+1f

0

]

=

[Kff 0

0 I

]−1([(ff )ext

0

]

−

[

(ff )jint

0

])

. (2.62)

The vector[Kfp∆dp 0

]−1which is necessary just for the first iteration can be computed

by first defining,[Kfp∆dp

Kpp∆dp

]

=

[Kff Kfp

Kpf Kpp

] [0

∆dp

]

(2.63)

and afterwards setting Kpp∆dp to zero.

Arc-Length Method

The essential difference between Arc-Length and load control method is that load factor λi ateach solution step is not prescribed. Instead, it is treated as an unknown which is defined bythe constraint equation for a given arc-length ∆s as shown in Fig.2.2.d. Hence, the nonlinearsystem of equations (2.52) is extended with an extra condition f(u, λ) = 0

[G(u, λ)f(u, λ)

]

= 0. (2.64)

Different arc-length methods vary by the constraint condition, e.g. Riks, and Crisfeld. The

above system should be linearized and then solved to obtain the vector of unknowns[u λ

]−1.

This is not further discussed here. For more information, also other constraint conditions,interested reader may refer to [Wri08].

21

Chapter 3

Review of Nonlinear Beam

Elements and Contact Formulations

So far, the fundamentals of continuum mechanics and FEA are treated in a general manner.In this chapter, theories that specially serve for the final purpose of this work, simulationof robot cables and their contact interactions with robot arms, are shortly reviewed. First,a review of existing beam theories and finite element beam elements is provided. Specialconsiderations that one has to take into account for nonlinear beam elements are also tackled.In the second section, a short overview of classical frictionless contact mechanics and variousmethods for contact regularization as well as contact discretization are presented. Finally, areview of research in the beam contact field is provided.

3.1 Nonlinear Beam Elements

A beam is a structural element that in one dimension is larger than the two others, henceprimary noted as a 1D element. A beam can resist lateral loads applied perpendicular toits axis, as opposed to truss that can be loaded only in axial mode. While there exist beamtheories that can model in-plane as well as out-of-plane deformations in the beam crosssection, most commonly used beam models assume a rigid cross section. Here, such beammodels are considered.

The well-known Euler-Bernoulli and Timoshenko theories are applicable to beams in the linearregime. While the former is a shear-free model, the latter can account for shear deformations[Tim21]. In 1859, Kirchhoff extended the work of Euler and Bernoulli to nonlinear deforma-tions in 3D space and initially curved configurations [Kir59]. Small axial tensions were addedto Kirchhoff theory by Love [Lov44]. However, the resulted Kirchhoff-Love beam model doesnot account for shear deformations. In two subsequent works [Rei72] and [Rei81], Reiss-ner added shear deformations to the theory while considering some approximations, whichwere finally treated by Simo [Sim85]. The resulting theory, Simo-Reissner beam model, isconsistent with 3D continuum kinetic and kinematic assumptions, and therefore is termedas Geometrically Exact (GE). The only restriction of GE beam theory is the assumption ofundeformed cross-sections. Strains can be finite and are consistent with the virtual workprinciple regardless of the magnitude of deformations. Since this is a shear-deformable the-

22 3. Review of Nonlinear Beam Elements and Contact Formulations

ory, at each point of the beam centerline six DOFs are required, three of which describe thecenterline’s position and the other three parametrize rotation field which defines orientationof the beam’s cross-section.

In context of the FEM, there are various methods to discretize and model geometrically non-linear beams. Common methods are co-rotational, Absolute Nodal Coordinates (ANC), solidbeam elements e.g. [SKIS14], [BB79] and geometrically exact beam elements. While ANCand solid beam elements are based on 3D continua, the underlying theories of co-rotationaland geometrically exact beam elements are based on 1D continua. The co-rotational ap-proach, first introduced in [BH73] [Wem69], defines a local co-rotating reference frame ateach element, whose rotation defines rigid body rotation of the element. Furthermore, ele-ment deformations are described in the local frame, for which first- or second-order theoriescan be used. In [Kre09] a co-rotational nonlinear Euler-Bernoulli beam element formulationin 2D as well as 3D is thoroughly explained. Geometrically exact beam elements are based onSimo-Reissner theory. In [Cri97] a historic review of GE beam element developments is given.A crucial step in GE beams formulation is discretization of the rotation field. In [ACJ99]and [JC99], a rotation interpolation was proposed that could preserve objectivity, observer orframe invariance, and path-independence of the GE element for the first time. As argued in[MPW17] this type of beam elements is state-of-the-art method for modeling geometricallynonlinear beam problems.

For the case of geometrically nonlinear and shear-free beams, besides some analytic treat-ments of Kirchhoff-Love theory [LS96] and [SH94], co-rotational shear-free beams are pro-posed e.g. in [LBH14]. However, this type of elements suffer poor accuracy [AV12] whichcan be due to lack of exact representation of kinetics and kinematics. This fact justifies theconcept of geometrically exact beams based on Kirchhoff-Love theory. The first 3D and largedeformation versions of such element were proposed by Boyer [BP04] and Weiss [Wei02a],[Wei02b]. These formulations assume isotropic cross sections and straight initial configura-tion. The work in [BDNLV11] is an extension to the above-mentioned [BP04] for modelingof undersea cables. Greco et al. developed an isogeometric element with anisotropic crosssection and initially curved configuration [GC13], [GC16]. But, this method was not appliedto nonlinear examples until the recent work of Bauer et al. [BBP+16]. Geometrically exactFEM beam elements of Kirchhoff type with anisotropic cross section and initially curvedconfiguration is formulated by Meier et al. [MPW14]. In a subsequent work [MPW15], areduced torsion-free model is derived that is suitable for modeling cable-like structures. Afurther issue of shear-free beams that should be taken into account is membrane locking,more discussed in the subsequent section. This issue is treated in [Mei16, MPW17].

In this work the torsion-free GE beam element of Kirchhoff type developed by Meier et al.is employed for modeling the deformations of robot wiring.

Membrane Locking in Beam Elements

From a mechanical point of view, locking in general can be explained by occurrence of par-asitic stresses in numerical solution that do not occur in exact analytic solution. It can bedistinguished by deterioration of spatial convergence rate in dependence of a critical parame-ter (e.g. element slenderness ratio ξele = lele/Rele in beams). Various types of locking effectsthat are observed in different structural elements are transverse shear, volumetric, shear, andmembrane locking [WBB16]. In geometrically exact beam formulations, the relevant lock-

3.2. Contact Mechanics 23

ing effects are shear locking and membrane locking. While shear locking is associated withSimo-Reissner shear-deformable finite element, shear-free Kirchhoff type beams are prone tomembrane locking.

Membrane locking, which was first introduced by Stolarski and Belytschko [SB82], is observedin curved elements such as beams and shells. It occurs due to the inability of elements toexactly reproduce inextensibility for curved configuration. In the case of beam elements, whena beam is imposed to constant bending moment, normal elongation and forces in the beamare non-zero. This can be traced back to the fact that trigonometric functions, which appearin pure bending analytic solution, cannot be exactly represented by shape functions. Here,membrane stiffness enlarges small errors in kinematic representation of beam deformationresulting in parasitic axial stresses.

In [MPW15] membrane locking phenomenon in Kirchhoff type geometrically exact beams isstudied. Accordingly, parasitic axial stress strain energy increases with the beam slendernessratio. Possible remedies for this locking effect are Assumed Natural Strains(ANS), ReducedIntegration (RI), and Minimally Constrained Strains(MCS). (See e.g. [HTK77], [NP81], and[MPW15])

3.2 Contact Mechanics

This section gives an overview of frictionless contact mechanics, also termed as normal con-tact. Since active parts of contacting bodies are not known in advance, it should be defined inan iterative manner. Hence contact problems are considered nonlinear unless sub-regions aretied together, which is the case for node-to-node contact discretization. In Sec. 3.2.1 severalcontact discretization methods are discussed. For the remainder of this section, we assumea contact element composed of a unique pair of discretizations of bodies under considerationwhich are potentially in contact. Each side of the element is defined as either a slave ora master surface side, where the slave side is the domain of numerical integration for thecontact contribution.

Depending on the discretization method, the minimum distance between a number of discretepoints on the master surface and the slave side are obtained via Closest Point Projection(CPP) [KS07] of the points onto the master surface. The discrete distance values are assignedto the gap function g, where negative gaps denote penetration of two bodies. Hence, contactintroduces a constraint to the problem which denotes,

g ≥ 0. (3.1)

At points where this constraint is violated, it should be enforced via a suitable regularizationtechnique (Sec. 3.2.2). Finally, in Sec. 3.2.3 some important aspects of beam contact arecovered.

3.2.1 Contact Discretization

In general, three categories of contact discretization techniques can be distinguished as.[Yas11], [Bog17]


• Node to Node(NTN)

• Node to Segment(NTS)

• Segment to Segment(STS).

The simplest discretization technique and the oldest one is Node-to-Node technique, whichis based on matching adjacent nodal positions in contacting bodies [FZ75]. Due to elementdistortion during large deformations, this method is only applicable for small deformations.Also, nodal sliding along the contact interface is not allowed.

Node-to-Segment [WVVS90] technique is applicable for non-matching meshes, large deforma-tions, and allows sliding of nodes along contact interface. Furthermore, it is a simple and ro-bust approach, which makes it an ideal method for contact implementation in general-purposefinite element codes. Gauss Point-to-Segment approach can be regarded as an extension ofNTS, in which contact constraint is evaluated at element Gauss points instead of nodes (seee.g. [Pop17]).

While both NTN and NTS methods evaluate contact constraint pointwise, Segment-to-Segment discretization technique, first introduced by Simo [SWT85], enforces the constraintin a weak manner (integration over contact interface). The mortar method is a special type ofSTS approach. For more information on this technique we refer to [Woh01], [FW05],[PGW09].

In the current work, contact interactions between the wiring and the robot linkage is modeledusing the Gauss Point-to-Segment approach while the robot linkage is considered as analyticsurface. Hence, the name Gauss Point-to-Analytic Surface (GPTAS) is employed to refer tothe utilized contact discretization technique.

3.2.2 Contact Regularization

As already discussed, contact introduces a constraint to the problem which is usually ofinequality type. As a result, a solid mechanics problem which includes contact effects can beviewed as a minimization of total potential energy π (2.19) subject to constraint (3.1),

minπ(φ) subject to g ≥ 0. (3.2)

This is similar to an optimization problem. For a mathematical treatment, We therefor referto Karush-Kuhn-Tucker (KKT) condition from optimization theory [LY08].

gN ≥ 0 (3.3a)

tN ≤ 0 (3.3b)

gN tN = 0. (3.3c)

Here, the subscript −N denotes normal direction and t is the traction force. While at thecontact interface either there is no penetration (3.3a) or a negative traction force applied tothe surface (3.3b) (when two contacting bodies are about to penetrate), two cases can becombined into one by (3.3c). Fig. 3.1.a is a graphical representation of this model.


gN

tN

gN

tN

ǫNgN

tN(a) (b) (c)

Figure 3.1: Contact regularization: normal traction force and normal gap distance (a) KKT Condi-tion, (b) Lagrange Multipliers Method, (c) Penalty Method

The KKT condition described above is for frictionless normal contact cases. Other contactconstitutive models are available that can account for other effects such as friction, adhesion,etc. There are various methods to enforce the KKT condition, among which Lagrange multi-pliers, Penalty, Augment Lagrange Multipliers, Barrier, and Nitsche’s method can be named.The choice of the regularization method depends on type of contact problem (e.g. beam or3D solid contact), the desired accuracy, and implementation complexity. Here, Penalty andLagrange multipliers methods are discussed. For other regularization methods, as well asother contact constitutive models we refer to [Wri06].

Lagrange Multipliers Method

The Lagrange multipliers method is a classical mathematical tool to add constraints to theweak form which includes additional variables to the problem (Lagrange multipliers λN ).This method is also the basis for connecting non-matching meshes in mortar discretizationtechnique introduced earlier.

The contact potential energy using λN can be formulated as

πLMc =

∫

Γc

(λN · gN )dA. (3.4)

The variation of the above potential is

δπLMc =

∫

Γc

λN · δgNdA+

∫

Γc

δλN · gNdA. (3.5)

Here, Lagrange multipliers λN can be defined as the contact traction force. The variationalform (3.5) consists of virtual work of contact forces (λN ·δgN ) and the enforcement of contactconstraint (δλN · gN ).

This method leads to exact enforcement of contact constraint (Fig. 3.1.a and Fig. 3.1.b).However, the resulting linear system of equations has a saddle point structure [Ble16]. Also,it ends up with a bigger number of unknowns due to the additional DOF λN at each node.


Penalty Method

In contrast to the Lagrange Multipliers method, the penalty method does not introduce newunknowns to the system. In this method, at points where contact constraint is violated(gN ≤ 0) a penalty force is applied. At such points, the contact potential reads

πpc =

1

2

∫

Γc

ǫN (gN )2dA, (3.6)

where ǫN is penalty parameter. Physically, it can be interpreted as the surface stiffness whichacts on points with negative gap values.

In penalty regularization, the contact constraint is exactly satisfied only when ǫN → ∞ (Fig.3.1.c). However, setting the penalty parameter to a very big number causes ill-conditioningof the stiffness matrix. In most cases, ǫN is set such that a trade-off between solutionaccuracy and computation time can be achieved. On the other hand, this method offers arelatively simple implementation. Also, contrary to the Lagrange multipliers method, positivedefiniteness of stiffness matrix is retained using Penalty regularization.

In Chapter 5, advantages and disadvantages of the two above-mentioned regularization tech-niques in beam contact applications are discussed.

3.2.3 Beam Contact

Contact of beams undergoing large displacement has many practical applications in bothclassical engineering and modern fields of research. Such applications include robot parts,woven fabrics (see e.g. [Dur10]), arterial stents in the biomedical engineering ([HK06]). Thecontact formulation discussed so far focuses on general solid contact modeling. Beams aredescribed as one-dimensional curves in 3D space. Consequently, their contact formulationrequires special treatments.

Wriggers and Zavarise [WZ97] proposed a formulation to model beam to beam contact inter-actions based on a discrete force applied at closest point between two beam centerlines. Theoriginal model considered circular cross-sections and frictionless contact, which can be referredto as point-to-point formulation. A detailed presentation on such formulation can be found in[Wri06]. This model is further extended to more general cases including frictional problems,adhesion effects, rectangular cross-sections, and self-contact. Neto et al. [NMP14, NPW13],proposed an extension for modeling contact interactions with rigid surfaces. Such beam-to-rigid surface contact interaction serves well for the purpose of this work. Nevertheless, forthe sake of completeness, further crucial beam-to-beam contact developments is discussed inthis section.

Point-to-point contact formulation is suitable for cases of beam-to-beam contact with largeintersection angle between two beams. On the other hand, for nearly parallel or entangledbeams (small contact angles) a distributed contact force might be more suitable, and denotedby line-to-line method. For this variant, formulations developed by Durville [Dur12] basedon a collection point-to-segment approach, or by Chamekh et al. [CMM09, CMM14] can bereferenced. Drawback of line-to-line contact methods is heavy computational effort in case


of large contact angles. Meier et al. [MWP17] suggested an all-angle beam contact methodthat combines advantages of both point- and line- formulations.

29

Chapter 4

Geometrically Exact Beam

As argued in Sec. 3.1, shear- and torsion- free beam elements are suitable models for cable-like structures. Among different beam element technologies, torsion-free Geometrically Exact(GE) beam of Kirchhoff-type seems to have advantages in both accuracy and computationalefficiency. In this chapter, this type of beam element based on the work of Meier [Mei16] isdiscussed.

Simo-Reissner beam theory which lays the foundation for the subsequent derivations is brieflydiscussed in Sec.4.1, followed by the enforcement of Kirchhoff constraints and the derivationof a simplified model of high practical relevance for modeling cables. Subsequently, in Sec.4.3 the FEM procedure is applied to the continuum model in order to derive the beamformulation, which is verified with numerical benchmarks in the final section of this chapter(Sec. 4.4)

4.1 Geometrically Exact Simo-Reissner Beam Theory

The GE Kirchhoff-type beam and the reduced torsion-free version are based on the shear-deformable Simo-Reissner theory. In this section, a summary of this theory is discussed. Formore detailed derivation, the interested reader is referred to the original works [Rei72, Rei81,Sim85] or [MPW14] instead.

For beam kinematics, we assume two right-handed Cartesian frames Ei and ei, i ∈ 1, 2, 3,where the former is the beam’s material configuration and the latter is its spatial config-uration. However, for simplicity we assume the mentioned configurations coincide. Thebeam’s centerline at initial configuration with length L is denoted by r0(s) ∈ ℜ3, wheres ∈ 0, L is arc-length parametrization of the curve. Furthermore, a field of orthonormaltriads s → g01(s),g02(s),g03(s) ∈ ℜ3, attached at each centerline point, defines the orienta-tion of the cross-section. The base vector g01(s) is tangent to the beam’s centerline,

g01(s) ≡ r′0(s) with ‖r′0(s)‖ = 1, (4.1)

where, ()′ = dds() denotes the derivative with respect to the arc-length parameter s, and

‖(...)‖ is the Euclidean norm. Base vectors g02(s) and g03(s) are defined in the direction

30 4. Geometrically Exact Beam

sg01

g02

g03

e1

e2

e3

g1

g2

g3

s

r0(s) r(s)

(a) (b)

Figure 4.1: (a) Initial and (b) deformed configuration of Geometrically Exact shear-deformable beamtheory and kinematic quantities

of the cross-section’s principal axes. The orthogonal transformation s → Λ0(s) ∈ SO(3)describes transformation from the global frame Ei to g0i.

g0i = Λ0(s)Ei(s) (4.2)

Here, SO(3) ∈ ℜ3×3 denotes the group of orthogonal transformations.

In a similar fashion, the deformed configuration is identified by the cross-section’s centroidr(s) ∈ ℜ3 and the triad g1(s),g2(s),g3(s) ∈ ℜ3 defining the cross-section’s orientation.Nevertheless, it should be noted that s is the arc-length parameter of the initial configuration,and hence ‖r′(s)‖ = 1 does not hold anymore. Base vectors g2(s),g3(s) are defined bydirection of cross-section principal axes. g1(s) is

g1(s) = g2(s)× g3(s), (4.3)

and therefore, perpendicular to the cross-section. But, due to the shear deformations, itis not tangent to centerline r(s). Orthogonal transformation tensor s → Λ(s) ∈ SO(3) isformulated similar to (4.2) as,

gi = Λ(s)Ei(s). (4.4)

Fig. 4.1 illustrates kinematic quantities essential in the definition of GE shear-deformablebeams.

Later in this section, derivation of base vectors gi(s) with respect to s, g′i(s), and their

variation δg(s) are necessary for derivation of the weak form of equilibrium and the deforma-tion measure. Using special rotation vector θ(s) and curvature vector k(s), the mentionedquantities are defined as [Sim85],

g′i(s) = k(s)× gi(s) (4.5a)

δgi(s) = δθ(s)× gi(s). (4.5b)

For derivation of the weak form, we start with balance of forces and moments equations whichcan be formulated as

4.2. Geometrically Exact Kirchhoff Beam Theory 31

f ′ + f + fρ = 0 (4.6a)

m′ + r′ × f + m+mρ = 0, (4.6b)

where, f and m are force and moment stress resultants. Also, f , m and fρ , mρ denotedistributed body and inertia forces and moments respectively. Here, the static problem isconsidered and as a result inertia effects are neglected.

With variation of displacement δr(s) and rotation δθ(s), the weak form of equilibrium derivedfrom (4.6) as

G =

∫ l

0[δθ

′T

m+(δr′−δθ×r′)Tf ]ds−∫ l

0[δθT m+δrT f ]ds−[δrTfσ+δθTmσ]Γσ = 0. (4.7)

We denote work-conjugated strains the stress measures m and f by γ and ω respectively(see Sec.2.1.3). Hence, the variation of the mentioned strain measures is defined by the weakform (4.7) as

δγ = δθ × r′ (4.8a)

δω = δθ. (4.8b)

The corresponding strain measures to the above variational forms are

γ = r′ − g1 (4.9a)

ω = k −ΛΛT0 k0. (4.9b)

Finally, to calculate stress measures f and m with strain measures (4.9), the St. Vennant-Krichhoff material law introduced in Sec. 2.1.4 is employed. In [Mei16], the 1D equivalentto the mentioned constitutive law is derived. Accordingly, it reads:

m = CM .ω, CM = diag[GIT , EI2, EI3] (4.10a)

f = CF .γ, CF = diag[EA,GA2, GA3]. (4.10b)

Hear, CM and CF are material constitutive tensors. diag[..., ..., ...] is a diagonal 3×3 matrix.Quantities E and G are Young and shear modulus, IT , I2, I3 are beam cross-section polar andprincipal moments of inertial, and A, A2, A3 are cross-section area and two reduced areas.

4.2 Geometrically Exact Kirchhoff Beam Theory

While geometrically exact Simo Reissner theory needs pointwise six degrees of freedom todescribe beam configuration (three translational and three rotational quantities), Kirchhoffbeams need only four degrees of freedom by assuming vanishing shear strains. This as-sumption is especially sensible for slender beams for which shear can be neglected. Degreesof freedom in Kirchhoff theory correspond to three translational components and one de-


scribing torsion of the cross section. Shear strains are neglected by Kirchhoff constraints.According to which, unique vectors g2(s) and g3(s) of the coordinate frame defined at thecross section remain perpendicular to the centerline tangent t(s), (t(s) = r′(s)):

g2(s).t(s) = 0, g3(s).t(s) = 0 (4.11a)

or

g1(s) =t(s)

‖t(s)‖. (4.11b)

Geometrically exact beam theory of Kirchhoff-type is based on enforcing this constraintto the Simo-Reissner theory(Sec. 4.1). Two approaches which are developed in [MPW14]and [MPW17] are Lagrange multipliers method or using a new parameterization based onthe Kirchhoff theory. In the former, the same parametrization as in the Reissner theory(r(s),Λ(s)) is employed to described the beam kinematics. To enforce the two constraints in(4.11a) two additional fields of Lagrange multipliers λ2(s) and λ3(s) are necessary which resultin a weak constraint enforcement. Alternatively, a new parametrization field (r(s), φ(s))which fulfills the Kirchhoff constraint in a strong manner (point-wise) can be used to describethe beam kinematics. Here, φ(s) describes torsional angle of cross-section with respect to thetangent vector t(s).

In ([MPW17]), the two approaches are compared in term of their accuracy and nonlinearsolver behavior. For moderately slender beams, both methods show comparable performance.On the other hand, the derivation of a torsion free beam, which is discussed in the nextsection, is more straightforward for the strong formulation. Hence, this approach is chosenas the Kirchhoff beam variant in the current work.

In the following, first the general geometrically exact beam theory of Kirchhoff-type, basedon strong Kirchhoff constraint fulfillment, is presented. This theory can account for torsiondeformations as well as non-isotropic cross-sections and initially curved beams. Afterwards, areduced formulation is treated in which an initially straight beam with isotropic cross-sectionand negligible torsion deformations is assumed. This version is of high practical relevance formodeling cable deformations, and therefore is used in this work.

4.2.1 Torsion Deformable GE Kirchhoff Beam

In the strong formulation, a new parameterization is employed which uses four DOFs in orderto define beam configuration at each point. The parameters the position vector of the beam’scenterline r(s) ∈ ℜ3 and cross-section’s angle with respect to centerline’s tangent φ(s) ∈ ℜ.

In order to define the orientation triad gi(s) of the beam’s cross-section, first a reference triadfield is defined as

gref,1(s) =t(s)

‖t(s)‖,

gref,2(s) = f2(r(s), r′(s), r”(s)),

gref,3(s) = f3(r(s), r′(s), r”(s)).

(4.12)


The base vector gref,1(s) of the reference triad is in the direction of beam tangent vectort(s). The other two remaining base vectors gref,2(s) and gref,3(s) are defined by arbitraryfunctions f2 and f3, which are only functions of the beam’s centerline r(s) and its derivatives.In [MPW17], two possible choices for f2 and f3 are discussed. However, the beam theorydoes not depend on the definition of these functions.

In the next step, the beam’s material frame gi(s) is generated by rotation of the referencetriad gref,i(s) around tangent vector t(s) by certain angle φ(s) as

g1(s) = gref,1(s),

g2(s) = gref,2(s)cosφ(s) + gref,3(s)sinφ(s),

g3(s) = gref,3(s)cosφ(s)− gref,2(s)sinφ(s).

(4.13)

Next, for deriving a Krichhoff-type beam, shear forces, which do not have any work contri-bution to the weak form, should be excluded from strong form (4.6). To this end, we splitthe force vector f into parallel and perpendicular components,

f = f‖ + f⊥ = f‖g1 + f⊥ (4.14)

By eliminating f⊥ from the strong form [MPW14], a system of four differential equationsdescribing GE Kirchhoff beam results:

(f‖g1)′ +

[r′

‖r′‖2× (m′ + m)

]′= 0 (4.15a)

gT1

(m′ + m

)= 0 (4.15b)

In order to derive the weak form, variational forms of the beam’s centerline position androtation vectors (δr(s), δθ(s)) are necessary. To calculate admissible variations of rotationvector which are consistent with Kirchhoff constraints, we split it into perpendicular andparallel components (δθ⊥ and δθ‖):

δθ = δθ‖ + δθ⊥ = δαg1 + δθ⊥ (4.16)

The parallel variational component α is not constrained by Kirchhoff conditions. There is anunderlying difference between two rotational parameters φ used in Kirchhoff parameterizationand α, which introduced here. In contrast to φ, which defines rotation with respect to thereference triad, α denotes the total torsion.

To calculate the perpendicular component of rotation vectors variation (δθ⊥), the variationalform of triads already expressed by (4.5b) can be rewritten as

δg1 = δθ × t = δθ⊥ × g1. (4.17)

Using the permutation rule of orthogonal cross product, δθ⊥ is calculated as


δθ⊥ = g1 × δg1 =r′

‖r′‖× δ

(r′

‖r′‖

)

=r′ × δr′

‖r′‖2. (4.18)

Finally, using (4.16)and (4.18) we can write an expression for the rotation vector variationas

δθ = δαt +r′ × δr′

‖r′‖2. (4.19)

By multiplying equilibrium of forces and moments with variations of beam centerline androtation vector, and after applying the integration by parts rule twice, the weak form, G, ofGE Kirchhoff beam results:

G(r, φ, δr, δα) =

∫ L

0

[

δθ′Tm+ δr′T g1f‖ − δrT f − δθT m]

ds−[δrTfσ − δθTmσ

]

Γσ= 0.

(4.20)

Together with the above weak form the problem setup is completed with boundary conditionsthat fulfill Neumann and Drichlet boundaries. These BCs are

on Γu: r = ru , g1 = g1u , α = αu ,on Γσ: f = fσ , m = mσ ,while Γu ∩ Γσ = ∅ and Γu ∪ Γσ = 0, L .

Objective deformation measures that are conjugate to stress resultants are derived in [MPW14].The resulting bending strain vector ω ∈ ℜ3 and tension (axial) strain ǫ ∈ ℜ are

ω =

τ + φ′ − τ0 − φ′0

(g2 · κ)− (g02 · κ0)(g3 · κ)− (g03 · κ0)

(4.21a)

ǫ = ‖r′‖ − 1. (4.21b)

Here, τ is torsion resulting from the reference triad , gref,i and can be calculated fromτ = g′

ref,2.gref,3. Additionally, κ is defined as Frenent-Serret curvature,

κ =r′ × r′′

‖r′‖2. (4.22)

In (4.21) terms with (...)0 are related to initial configuration of the beam. In case of aninitially straight beam, these terms can be omitted.

Expressing stress resultants with the help of a constitutive law is final step of the beamderivation. Again, the similar constitutive St. Vennant Kirchhoff relations used in Simo-Reissner beam derivation 4.1, shall be used here,

f‖ = EAǫ,m = CMω (4.23)


4.2.2 Torsion Free GE Kirchhoff Beam

As discussed in the last section, general theory of GE Kirchhoff beam considers torsion.The assumption results in an extra degree of freedom φ. Next to various advantages thisassumption has in many applications, there are a few drawbacks. Firstly, in order to define φat each point, we have to establish a reference triad and update it at each solution iteration.The triads also must be consistently linearized in case a gradient based solver is used to solvethe nonlinear weak form, which adds more complexity and computational effort. In addition,the linearization of δθ‖ component of rotational vector variation, which showed up in (4.16)due to considering torsion, causes a non-symmetric stiffness matrix. Solving non-symmetricsystem of equations is even more computational expensive. Hence, a torsion-free version ofGE Kirchhoff beam is more numerically efficient.

In [MPW15], the torsion-free model is presented by canceling out the degree of freedom φfrom the formulation. It is proved that this model is still “exact” since it is consistent withthe Kirchhoff constraint. Additionally, the authors argue that this model is more stable whenis used with cable-like structures in comparison with torsion-deformable formulations, wherelow torsional stiffness of cables causes ill-conditioning of the stiffness matrix, or pure cableformulations, where artificial bending is often necessary in order to stabilize compressionalmodes.

In addition to resigning the torsion-related degree of freedom from GE Kirchhoff beam for-mulation, for the final purpose of this work we can assume so called “isotropic bending”problem case, with an initially straight beam with quasi-circular cross sections:

I2 = I3,κ0 = 0. (4.24)

Inserting the mentioned assumptions, isotropic bending and vanishing shear deformation,into (4.21) and (4.23) bending strain and stress resultants are simplified as

ω = κ (4.25a)

m = EIκ. (4.25b)

Excluding torsion from beam formulation, torsion contribution to the variation of rotationvector (4.16) (the parallel component) should be removed, hence

δθ = δθ⊥ =r′ × δr′

‖r′‖2. (4.26)

Using the tweaked rotation vector’s variation, the weak formulation, G, can be expressed as:

G(r, δr) =

∫ L

0

[

δθ′Tm+ δr′T g1f‖ − δrT f − δθT m]

ds−[δrTfσ − δθTmσ

]

Γσ= 0 (4.27)


Although weak formulations (4.20) and (4.27) have the same explicit formulas, it should benoted that unlike torsion deformable formulation, the torsion-free weak form is no longer afunction of torsional rotation angle φ and its variation δα.

4.3 Geometrically Exact Kirchhoff Beam Element

In Sec.2.2, the general procedure of nonlinear finite element formulations is discussed. In thiswork, the used beam element is based on the shear-free weak formulation. In the following,first, the weak form is discretized. Afterwards, linearization and numerical integration aretreated. While issues of path-independency and objectivity of the derived formulation arenot treated here, the interested reader is referred to [MPW17] for a rigorous mathematicalproof and numerical examination on these issues.

4.3.1 Finite Element Discretization

In this section, the discretization of the primary variable r and its variation δr is handled. Thehighest derivative of beam’s centerline r(s) in the mentioned weak form is r”(s). Hence, fol-lowing the compatibility condition, the employed shape functions should be C1−continuous.In the original paper [MPW14], C1−continuous Hermite shape functions of order three areused which lead to two-noded elements with discretized initial and deformed form as,

r0,h(ξ) =

2∑

i=1

H id(ξ)d

i0 +

c

2

2∑

i=1

H it(ξ)t

i0 (4.28a)

rh(ξ) =2∑

i=1

H id(ξ)d

i +c

2

2∑

i=1

H it(ξ)t

i. (4.28b)

In this work, subscript (...)h of a quantity denotes its discrete representation. Also in (4.28),vectors d and t are nodal positions and tangents, scalar value quantity c is a constant whichwill be defined in the following, and the functions H i

d(ξ),Hit(ξ) are Hermite polynomials with

parametric space ξ ∈ [−1,+1] defined as (see also Fig. 4.2) ,

H1d (ξ) =

1

4(2 + ξ)(1− ξ)2 (4.29a)

H2d (ξ) =

1

4(2− ξ)(1 + ξ)2 (4.29b)

H1t (ξ) =

1

4(1 + ξ)(1− ξ)2 (4.29c)

H2t (ξ) = −

1

4(1− ξ)(1 + ξ)2 (4.29d)

4.3. Geometrically Exact Kirchhoff Beam Element 37

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0

0.5

1

ξ

H(ξ)

H1d(ξ)

H2d(ξ)

H1t (ξ)

H2t (ξ)

Figure 4.2: Hermite polynomials

Substituting nodal values of parametric space in (4.29) , ξ1 = −1 and ξ2 = +1, it is straight-forward to show that the mentioned polynomial fulfills the interpolation property conditionof shape functions. Furthermore, in [MPW14] completeness of this shape functions for allpolynomials up to order three is proved.

As shown in (2.39), discretization of δr has the same form as rh, thus,

δrh(ξ) =2∑

i=1

H id(ξ)δd

i +c

2

2∑

i=1

H it(ξ)δt

i (4.30)

In the following, the discretization of r′0(s) and r′(s) which appear in the weak form is alsotreated. Considering first line of (4.28a) and discrete differential of arc-length s, dsh =‖r0,h,ξ(ξ)‖dξ, and using chain derivative rule, r′0,h(ξ) is

r′0,h(ξ) = r0,h,ξdξ

dsh=

1

‖r0,h,ξ(ξ)‖(

2∑

i=1

Hd,ξ(ξ)di0 +

c

2

2∑

i=1

H it,ξ(ξ)t

i0)

=1

J(ξ)r0,h,ξ(ξ)

, (4.31)

where, J(ξ) = ‖r0,h,ξ(ξ)‖.

In this work, (...),ξ denotes first derivative with respect to ξ, and (...),ξξ is second derivative .

Similarly, the discretized form of r′(s) is

r′h(ξ) =1

J(ξ)(

2∑

i=1

Hd,ξ(ξ)di0 +

c

2

2∑

i=1

H it,ξ(ξ)t

i0)

=1

J(ξ)rh,ξ(ξ)

. (4.32)

The second derivative of beam centerline r(s) with respect to s, which is necessary for cal-culation of curvature κ, can be derived using the same concept as in r′0,h,ξ(ξ),

r′′h(ξ) =1

J(ξ)2rh,ξξ(ξ)−

J,ξ(ξ)

J(ξ)3rh,ξ(ξ). (4.33)


For the choice of constant c in Hermite polynomials (4.29), [MPW14] discusses four differentpossibilities that can affect discretization accuracy. However, it is shown that at the limitof fine discretization all the methods coincide. Additionally, all the choices guarantee theoptimal spatial convergence rate for h−refinement. As a result, the method with the leastcomputational cost is adapted here. Accordingly, c is set to the element initial length as

c = ‖d10 − d2

0‖. (4.34)

For a simpler discretization notation, element-wise assembly of shape functions H and nodalpositions and tangent vectors d is adapted. The assemblies read as

H ≡

(

H1d (ξ)I3,

1

2cH1

t (ξ)I3,H2d (ξ)I3,

1

2cH2

t (ξ)I3

)

(4.35a)

d ≡(

d1T , t1T , d2T , t2T)T

(4.35b)

where, I3 denotes the identity matrix of order three. Discretizations (4.28) and (4.30) canbe rewritten in a short format as

r0,h(ξ) = Hd0 rh(ξ) = Hd (4.36)

δrh(ξ) = Hδd. (4.37)

In a similar fashion, discrete derivatives of r and δr are,

r′0,h(ξ) = H ′d0 r′h(ξ) = H ′d r′′h(ξ) = H ′′d (4.38)

δr′h(ξ) = H ′δd (4.39)

Also, the discrete form of the spin vector’s variation (4.26) should be determined. To thisend, we introduce S(a) ∈ so(3) with properties S(a)b = a× b and S(a)T = −S(a)T , wherea, b ∈ ℜ3 and (..)T represents transpose of a matrix. Additionally, so(3) denotes set of skew-symmetric tensors. Vector a is so-called axial vector of tensor S(a). Also, considering vectort = r

′

‖r′‖2 , relation (4.26) can be rewritten as

δθ =S(r′)δr′

‖r′‖2= S(t)δr′. (4.40)

Next, the discrete variation of the spin vector and its derivative can be formulated as

δθTh = −δdTH ′TS(t) ≡ δdTVθ⊥ (4.41)

δθ′T = δdTV ′θ⊥, (4.42)

4.3. Geometrically Exact Kirchhoff Beam Element 39

where, Vθ⊥ = −H ′TS(t).

Finally, substituting the derived discrete quantities into the weak form (4.27), its discretizedcounterpart Gh results as

Gh(d, δd) = δdT

(∫ l

0

[

V ′θ⊥m+H ′T g1f‖ −HT f − Vθ⊥m

]

ds−[HTfσ + Vθ⊥mσ

]

Γσ

)

= 0.

(4.43)

The residual force vector r(d) can be formulated from the above discrete weak form as

r(d) =

∫ l

0

[

V ′θ⊥m+ Vǫf‖ −HT f − Vθ⊥m

]

ds−[HTfσ + Vθ⊥mσ

]

Γσ= 0. (4.44)

Here, Vǫ = H ′T g1 is used.

4.3.2 Linearization

If solution algorithms that require tangent information (e.g. Newton-Raphson solver) areemployed for solving the nonlinear system of equations, linearization of the residual forcevector (4.44) is necessary. In this section, we deal with this matter.

First, linearization of some definitions which already are used in beam theory derivation anddiscretization process are presented. These linearizations are necessary for further lineariza-tion of (4.44).

Remark 2. It should be noted, due to the similar mathematical concept of linearization andvariation, discrete linear form of r and its derivatives can be formulated by substituting deltaoperator (δ) with ∆ into the discrete variational forms (4.37) (4.39) already derived.

∆t =1

‖r′‖(I3 − 2g1 ⊗ gT

1 )H′∆d (4.45a)

∆t′ =−2(r′T r”)

‖r”‖4(I3 − 2g1 ⊗ gT

1 )H′∆d

−2

‖r′‖2(g′

1 ⊗ gT1 + g1 ⊗ g′T

1 )H ′∆d

+1

‖r′‖2(I3 − 2g1 ⊗ gT

1 )H′′∆d

(4.45b)

∆g1 =1

‖r′‖(I3 − g1 ⊗ gT

1 )H′∆d (4.45c)

The linearized form of residual forces (4.44) has the general form

∆r(d) =

∫ l

0

[∆V ′

θ⊥m+ V ′

θ⊥∆m+∆Vǫf‖ + Vǫ∆f‖ −∆Vθ⊥m

]ds− [∆Vθ⊥mσ]Γσ

. (4.46)


Considering (4.22) and (4.25b), the term ∆m in the above equation can be written as (em-ploying the concept of skew-symmetric tensor S(a) ∈ so(3) used in the last section)

∆m = EI∆κ = EI(−S(r′′)∆t+ S(t)∆r′′). (4.47)

Other linear forms related to moment in (4.46) are

∆V ′θ⊥m = H ′′S(m)∆t+H ′S(m)∆t′ (4.48a)

∆Vθ⊥mσ = H ′TS(mσ)∆t. (4.48b)

The distributed moment related term ∆Vθ⊥m has the same form as second line of the aboveequation.

Furthermore, formulation of terms ∆f‖ and ∆Vǫ is,

∆Vǫ = H ′T∆g1 (4.49a)

∆f‖ = EA∆ǫ = EAV Tǫ ∆d (4.49b)

In order to calculate tangent stiffness matrix Ktan, the vector of nodal primary value incre-ments ∆d should be factorized from linearization (4.46) which results in ∆r = Ktan∆d.


In this section, the integration of the stiffness matrix (4.46) and residual forces (4.44) isperformed by the Gauss quadrature rule (2.45). Both integral cannot be exactly calculatedusing Gauss rule there exist some terms in the expressions that are not purely polynomial.In [MPW14] however, via a set of numerical studies, it is shown that there is no remarkabledifference in integral result for Gauss points nG ≥ 4. Unless stated otherwise, nG = 6 Gausspoints are used in this work to perform integrations for GE Kirchhof beam stiffness matrixand residual force vectors.

For the following, consider f being the integrand in (4.46) or (4.44). According to (2.45),numerical integration of f reads

∫ l

0fds =

nGP∑

i=1

ωξif(ξi)J(ξi), (4.50)

where, Jacobian J is J(ξ) = ‖r0,h,ξ(ξ)‖. For the case of initially straight beam this expressionis simplified to J = l

2 .

4.4. Numerical Examples 41

4.4 Numerical Examples

Numerical examples introduced in this section are intended to investigate the performance ofthe presented beam element. In the first test case, for which an analytic solution exists thespatial convergence order of the beam is investigated. Results are in agreement with expectedconvergence order. Next, in the second example number of necessary Newton iterations andaccuracy of result, when a more complex loading is applied, are studied.

In example case with analytic solution relative L2− error (‖e‖2rel) is used, which reads

‖e‖2rel =1

umax

√

1

l

∫ l

0‖rh − rref‖2ds. (4.51)

In (4.51) rh, rref are numerical and reference (in this case analytic) solutions. Normal-ization with respect to element length l makes error measure independent from number ofelements. In [MPW17] optimal convergence rate for the presented beam element is discussed.Convergence rate of L2−error is [SF08]:

‖e‖2 = O(hk+1 + h2(k−m+1)) (4.52)

The highest arc length derivative in the weak form (4.20) is 2 (m = 2) and trial functions ofpolynomial degree 3 (k = 3) are used in sec.4.3.1. Additional variable h in (4.52) representsthe element length. Hence, the optimal convergence rate of 4 is expected for L2−relativeerror. In the subsequent examples, a beam with initial length L = 1000, and square crosssection with side length R,A = R2, I = R4/12, and Young’s modulus E = 1 is set.

4.4.1 Example 1: Pure bending in 2D

Test case used in this section is a well-established benchmark that is used to study theconvergence of a wide range of nonlinear beam elements, e.g. in [Kre09], [MPW14], [BBP+16].A beam is clamped at one end and loaded with a pure moment M in the other end (see Fig.4.3). If external bending moment is set to M = EIπ

2l , then its deformed shape is an exactquarter circle.

Remark 3. In cases where the orientation of a node i has to be predefined, e.g. clamped endin this example, orientation of the nodal tangent vector ti is necessary to be set. However,as shown by (4.21b), the magnitude of the tangent vector defines tension strain at the node,and hence should be calculated during solution. In this example, where nodal orientation atclamped end coincides with the material frame, the nodal tangent vector is

t0 = (t01, 0, 0)T , (4.53)

where, value of t01 is not prescribed, rather is part of the solution.

The simulation is repeated with 1, 2, 4, 8, 16, 32 and 64 elements. Additionally, in orderto investigate the membrane locking effect different slenderness ratios ξ = L

R= 10, 100, 1000

and 10000 are considered.


M

Figure 4.3: Pure being in 2D, a beam is clamped at one end and loaded with point moment M = EIπ

2l

at the other end

101 102 103 10410−12

10−9

10−6

10−3

100

h(element length)

‖e‖2 r

el

order 4ξ = 10ξ = 100ξ = 1000ξ = 10000

Figure 4.4: Moment applied to the end of a clamped beam with analytic solution, L2 relative errorvs. element length for different beam slenderness ratios


x

yx

x

y

x

z

y

z

+

+

fx

f⊥sinφ

f⊥cosφ

Figure 4.5: 3D Line loading, a beam is loaded with a 3-dimensional load, all the boundary DOFsare constrained except ux at the right end.

In Fig. 4.4 the relative L2−error with respect to the analytic quarter circle solution for thesecases is presented. For the beam with slenderness ratio ξ = 10 the optimal convergence orderof 4 is met. By increasing slenderness ratio, at coarse meshes the membrane locking effectcauses higher error. However, by using finer mesh this problem can be sidestepped. For arelatively fine mesh of 64 elements all the cases have almost the same relative error. This isdue to the fact that critical value for the observed locking effect is element slenderness ratio(ξel =

leleR) rather than beam slenderness ratio. For application in modeling wire deformations

with typical slenderness rations laying between 10 to 100 and acceptable error less than 3percent, we can observe no additional anti-locking methods are necessary to be used.

4.4.2 Example 2: 3D line loading

Second test case studies solution accuracy and performance of nonlinear solver when a com-plex load is applied. An initially straight beam with slenderness ratio ξ = 1000 is subjected

to a 3-dimensional load f(s) = (fx, f⊥sinφ, f⊥cosφ) withfxEI

= f⊥2EI

= 3.6.10−8 and φ = 2πsl

(Fig. 4.5). All the boundary degrees of freedom are constrained to zero except the displace-ment in x direction (ux) at the right side of the beam. Again similar to last example the beamis discretized with 1, 2, 4, 8, 16, 32 and 64 elements. Error is calculated at the beam middlepoint. Reference solution(taken from [MPW15]) is a very fine discretization(32768 elements)with Simo-Reissner beam elements, which are shear- and torsion- deformable (4.15),

erel(l/2) =|uh(l/2) − uref (l/2)|

‖uref (l/2)‖(4.54)

Fig. 4.6 illustrates the relative error at the beam midpoint. Error decreases up to almostthree orders by using finer mesh but afterwards it levels off. The remaining error couldbe caused due to the shear deformations. Shear deformations are considered in referencesolution, while GE Kirchhoff beam element is shear-free.


101 102 103 104

10−3

10−2

10−1

100

h(element length)

e rel(l/2)

ξ = 1000

O(h1.79)

Figure 4.6: 3D line load applied to beam with a numeric reference solution from [MPW15] error indisplacement at the beam middle point vs. element length

This case is solved within 40 load steps for all discretizations. Number of iterations in alldiscretizations except coarse ones with one and two elements, which also show poor solu-tion accuracy, was 120 which means 3 iterations per load step on average. Furthermore, theNewton-Raphson solver has the expected quadratic convergence in iterative solution proce-dure. This effect shows that the linearization of the beam element is consistent.

45

Chapter 5

Beam to Surface Contact

In chapter 3 introductory information about general contact mechanics and recent researcheson beam contact was reviewed. In this chapter, the topic of beam to analytic surface contact,which serves for modeling of contact interactions between cables and robot arms, is in focus.

In the contact algorithm, first, the subsections of the two bodies which are in contact haveto be identified. Usually, this is done in two steps. In the first step, in a so-called globalsearch phase, the potentially contacting segments are defined. Subsequently, in a secondmore accurate search, the local search, the contacting pairs are defined. However, in caseswhere the number of subsections in one of the bodies or both is limited, e.g. in contactwith analytic surfaces where one of the surfaces consists of just one section, the global searchmight be skipped. As discussed in Sec. 3.2, the two surfaces in contact are referred to asslave and master. In this chapter, a beam element with centerline r(s) is considered as theslave side. During the local search, integration points onto the slave side are projected tothe master surface using Closest Point Projection (CPP). This is further discussed in Sec.5.1.Then, using the integration point coordinates on the slave surface and the projected point onmaster side, the gap value is calculated. In order to prevent the two surfaces from contact,the gap should be constrained to non-negative values. In Sec. 5.2 two approaches to enforcenon-negative gap values, the Penalty and Lagrange Multipliers methods, are compared andadvantages of the penalty method in application of the beam to surface contact are presented.Integration points with penetration (negative gap distance) contribute to the tangent stiffnessmatrix and the internal forces vector. This method is referred to as Gauss Point To AnalyticalSurface (GPTAS) formulation. In Sec. 5.3, the penalty potential of penetrating Gauss Pointsand its finite element discretization are established. Finally, two numerical examples to verifythe introduced contact enforcement methodology are presented in Sec. 5.4.

5.1 Closest Point Projection

CPP projection can be formulated as the minimization of the distance, d, between a pointon the slave surface with parametric coordinate ξs, rs(ξs), and master surface xM (ξ)

dmin(ξ) = min‖rs − xM (ξ)‖. (5.1)

46 5. Beam to Surface Contact

e1

e2

e3

ξ1

ξ2

r(ξs)

xM (ξ)

p

xM,ξ1

xM

,ξ2

Γ

Figure 5.1: CPP of point P onto the surface Γ

In (5.1), ξ = (ξ1, ξ2) is parametrization of the master surface Γc,M . The final goal is tofind ξ corresponding to the minimum distance. This can be achieved by calculating zeros ofgradient of distance function between two surfaces [WZ97],

grad d =

(∂d

∂ξ1,∂d

∂ξ2

)

= 0. (5.2)

As long as the master surface is convex, we can make sure (5.2) has a unique solution foreach point rs on the beam [Wri06]. Computing the gradient of the distance function resultsin two orthogonality conditions

((rs(ξs)− xM (ξ)) · xM,ξ1

(ξ)

(rs(ξs)− xM (ξ)) · xM,ξ2(ξ)

)

= 0. (5.3)

From Fig. 5.1, one could visualize that at the CPP of rs(ξs), the distance vector between thetwo surfaces is normal to the master surface. As a result, this vector is orthogonal to surfacetangent vectors xM,ξ1

and xM,ξ2at xM (ξ).

For an arbitrary surface, the set of equations (5.3) is solved using the Newton-Raphsonnonlinear solver. However, for some analytic surfaces, the CPP of a point can be computedby simple closed form solutions. Additionally, for numerical surfaces, which mostly consistof linear triangular meshes, one could use the solution proposed by Heidrich [Hei05]. Thesesolutions are further discussed in Appendix A.

Having xM (ξ) corresponding with projection of rs(ξs) onto the master surface, the gapfunction between two bodies can be defined as

g = (rs − xM ) · nM −Rs. (5.4)

In (5.4), the scalar Rs is the beam radius and the vector nM is the normal to master surfaceat xM (ξ).

5.2. Penalty Method vs. Lagrange Multipliers 47

Remark 4. the orthogonality conditions (5.3) guarantee the perpendicularity of distancevector d(ξ) to the master surface. However, this vector is not necessarily perpendicular tothe beam’s centerline rs. Hence, the gap distance from master surface to the beam’s surface(5.4) is not exact. For an exact representation, the term Rs should be divided by the cosineof contact angle. Nevertheless, this is an approximation commonly used in practice in linecontact algorithms (e.g. in [Mei16]).

5.2 Penalty Method vs. Lagrange Multipliers

In Sec.3.2.2, the Lagrange multipliers and the penalty method as two common practice contactregularization techniques were introduced. While contact enforcement using the former isa standard practice in applications which deal with contact in solids, for beams contactusually the latter is preferred [Mei16]. This can be due to basic difference between solidsand 1D structures. Firstly, as already discussed, the Lagrange multipliers method adds extraunknowns to the system for each surface DOF. The ration of surface to total DOFs in solidsis smaller than the one in beams. As a result, computational efficiency deterioration dueto these extra unknowns is worse for beams. Additionally, since local deformations in 3Dsolids problems are of interest, the extra computational expense is acceptable. However, inbeams, primary interest is usually the global behavior of the beam rather than cross-sectiondeformations. Accordingly, in many beam models a rigid cross section is assumed. Hence,slight cross section penetrations caused in the penalty method are acceptable in beams.

5.3 Contact Enforcement Using Penalty Method

Once the gap function at each integration point is evaluated using (5.4), the non-penetrationcondition of contact constrains the gap to non-negative values. For negative values, we enforcethe non-penetration condition in the normal direction using the penalty method. The penaltyregularization method can be physically interpreted as considering a surface stiffness factorǫ for the contacting bodies. If the gap distance between the two bodies is negative, thenthe considered surface stiffness applies a load in the opposite direction to compensate forpenetration. In Sec. 5.3.1, the penalty contact potential and its weak form along a beam isintroduced. Then in Sec. 5.3.2 and 5.3.3, the finite element discretization and its linearizationare discussed.

5.3.1 Penalty Potential and weak form

The non-penetration condition constrains the gap distance to be

g > 0. (5.5)

Enforcement of the above constraint using the penalty potential along beam element withlength l reads


π =1

2ǫ

∫ l

0〈g〉2ds, (5.6)

where, the penalty parameter is ǫ and term 〈g〉 denotes

〈g〉 =

g g < 00 g > 0

. (5.7)

To consider the contact effect in the weak form of the problem, the variation of 5.6 is calculatedas

G = ǫ

∫ l

0〈g〉δgds, (5.8)

for which the variation of the gap function is necessary. Since the master surface is a rigidbody, variation of xM in (5.4) is zero. Additionally, current position of the beam’s centerlinereads

rs = Xs + us, (5.9)

where, the terms Xs and us are beam centerline initial position (with variation equal to zero)and its displacement, respectively. Thus, the variation of gap distance is

δg = (δrs) · nM = δus · nM . (5.10)

Considering the orthogonality conditions (5.3), the normal vector nM in (5.10) can be definedas

nM =rs − xM

‖rs − xM‖. (5.11)

5.3.2 Finite Element Discretization

In order to have a finite element model that can consider contact effects on beam deformation,we need to discretize the weak form (5.8). In Sec. 4.3.1 the displacement of the beam’scenterline and its variational form were discretized using Herimitian shape functions as

ush = Hd, δush = Hδd. (5.12)

In beam element formulation in Chap.4, Hermitian shape functions were used. However, thecontact formulation does not depend on the choice of shape functions and H can be denotedas matrix of shape functions used in beam formulation. Terms d and δd are the unknownnodal displacement vector and its variation.

Introducing the beam discretization (5.12) into the gap function variation leads to

δgh = (Hδd) · nM (5.13)

5.3. Contact Enforcement Using Penalty Method 49

Finally, we can compute the discretization of the contact weak form by inserting the abovediscrete form into (5.8), which results in

Gh(d, δd) = δdT ǫ

∫ l

0〈gh〉H

TnMds. (5.14)

Consequently, the contact residual force vector rc(d) is

rc(d) = ǫ

∫ l

0〈gh〉H

TnMds. (5.15)

5.3.3 Linearization

Since the final value of the gap function and the segments of the bodies which are in contactat the final configuration are not known prior to the solution, the contact problem renders asnonlinear even if we assume the slave and master bodies have small deformations and linearelastic material behavior. Consequently, a nonlinear Newton-Raphson solver is employed tosolve the nonlinear system. The linearized form of the contact weak form variation (5.8)follows as

∆G = ǫ

∫ l

0

[

〈g

|g|〉∆gδg + 〈g〉∆δg

]

ds. (5.16)

The linearization of the gap function has the same form as its variation. Hence, the linearizedcounterpart of (5.13) can be written as

∆gh = (H∆d) · nM = nTMH∆d. (5.17)

Additionally, ∆δg is necessary to complete the linearization (5.16). Konyukhov in [Kon11,Kon15] argues that major contribution to contact tangential matrix stems from terms relatedto ∆g rather than ∆δg. Furthermore, due to its complex form, the numerical computationof ∆δg is much more expensive comparing to the computation of ∆g. Hence, in this workwe shall neglect ∆δg in our linearization for the expense of a few more iterations with cheap∆g evaluations. Plugging (5.13) and (5.17) into (5.16), the contact linearized form can bederived as,

∆Gh(d, δd,∆d) = ǫ

∫ l

0

〈gh|gh|

〉δdTHTnMnTMH∆d+ 〈gh〉

:=0︷︸︸︷

∆δg

ds

= δd

[

ǫ

∫ l

0〈gh|gh|

〉HTnMnTMHds

]

∆d .

(5.18)

The linearization of the contact residual force vector (5.15) results by neglecting δd from theabove equation.

∆rc(d) = ǫ

∫ l

0〈gh|gh|

〉HTnMnTMHds (5.19)



The discrete contact residual forces and the tangent stiffness matrix derived in (5.15) and(5.19) are evaluated numerically using the Gauss quadrature rule with nGP integration pointshaving coordinate ξigp and weight ωigp as ,

∫ l

0C(ξigp)ds ≈

ngp∑

igp=1

C(ξigp)ωigpJ . (5.20)

The numerical integration (5.20) is evaluated for one beam element. The Jacobean J stemsfrom mapping element boundaries [0, l] to numerical integration domain [−1, 1] and can becalculated similar to Sec. 4.3.3. Furthermore, the integrand C(ξigp) is the discrete contactresidual or the tangent stiffness matrix.

The contact gap function is the main quantity of the integrand in (5.20). As already discussedin Sec.2.2.4 using the Gauss quadrature rule exact integration for polynomials up to orderp = 2ngp − 1 is guaranteed. However, the gap function is discontinuous, and hence cannotbe exactly integrated. In [MPW16], it is proposed to use more contact integration intervalsalong the beam element to avoid these discontinuities for beam to beam contact applications.In beam to analytic surface contact, this problem is less severe specially in contact cases withsmooth surfaces. As a result, our approach to deal with this problem in the current work isusing more integration points to have a trade-off between integration accuracy and numericalefficiency.

Finally, in order to consider contact effect in the beam deformation, we shall add contactresidual forces and tangent stiffness to the beam linear system of equations as ,

(KΩ +KC)∆d = fΩ,ext − fΩ,int − fC,int. (5.21)

Here, terms with subscript Ω and C denote the beam element and contact contributions,respectively.

5.4 Numerical Examples

To study the numerical accuracy of the discussed contact formulation, two numerical examplesare treated in this section. In the first example, a similar version to the normal contact patchtest for beams is presented to investigate the error in the employed gap function’s numericalintegration. In the second example, contact between a beam and a rigid cylinder is modeled.Under a specific configuration for the contact penalty function and boundary conditions,an analytic solution to this problem exists. Hence, the relative L2−error (4.51) is used toinvestigate the spatial convergence behavior.

5.4.1 Example 1: Beam Patch Test

The contact patch test proposed by Taylor and Papadopoulos [TP91] is a standard benchmarktest for 2D and 3D contact element formulations, in which two contacting bodies are pressed


ǫ

p

Figure 5.2: Beam patch test, a beam is pressed against a rigid plane with constant pressure

against each other with constant pressure. This test is performed to study whether thecontact algorithm is capable to transfer constant pressure via the contact interface. In asimilar fashion, a beam is pressed against a rigid plane with a constant line load in thisexample.

Fig.5.2 shows the problem case setup. Beam’s geometric and mechanical properties are:length l = 1000, circular cross section with R = 10, and Young’s modulus E = 1 · 109. Theconstant line load applied to the beam is f = (0,−p, 0) with p = 1. And the chosen penaltyparameter for contact is ǫ = 1 · 106.

The contact penalty parameter is depicted in Fig.5.2 as its physical equivalent, a line spring.Based on this fact, the contact force can be easily calculated as fC = ǫ · g. Hence, a constantgap value along the beam results a constant contact force. The exact solution for the gapfunction is

gref = −p

ǫ. (5.22)

The problem is simulated for different number of integration points per contact element. Fig.5.3 shows the maximum error in the gap at integration points. The error in the gap functionis calculated as

e =‖gh − gref‖

‖gref‖. (5.23)

Here, we can conclude that at least two Gauss points are necessary in order to integrate aconstant gap function (5.22) exactly. For higher order gap functions more integration pointswould be necessary. In this work, unless stated otherwise, the same number of Gauss pointsused in the beam element (6 for third order Kirchhoff-type GE beams) is utilized for contactelement as well.

Remark 5. In this example, the beam can have rigid body movements in Y−direction. Hence,the resulting tangent stiffness matrix is singular. However, this holds only for the first iter-ation at the first time step, when contact forces are still not activated. Afterwards, due topenetration in the master surface, a contact load is applied to the beam, which in turn hindersthe rigid body movement and the tangent stiffness matrix is no longer singular.

5.4.2 Example 2: Twisting of a Beam Against a Rigid Cylinder

A more complex beam contact case is studied in this example. An initially straight beam withlength lbeam = 5.0, circular cross section Rbeam = 0.01, and Young’s Moduli E = 1.0 ·109 is incontact with a rigid cylinder with the same radius and length (Rcylinder = 0.01, lcylinder = 5.0)and a parallel centerline as illustrated in Fig.5.4. The beam is clamped at one end, and thecenter point of the cross section on the other end is moved on a circular path.


1 2 3 4

0

2

4

6

·10−2

number of Gauss points

max

errorin

gapdistance

Figure 5.3: contact patch test

x

y

z

cylinder

beam

g0

Figure 5.4: Twist of a beam around a rigid cylinder, initial configuration


In [Mei16], an analytic solution for a rather similar problem, the contact of two parallelbeams, is derived. The contact penalty parameter and boundary conditions are set such thatbeam’s deformed centerline is an analytic helix with constant slope h and radius r with thefollowing parametrization as

r(φ) =

hφr cosφr sinφ

, φ ∈ [0, 2π], r = Rbeam +Rcylinder − |g0|

h =

√(

l

2π

)2

− r2

. (5.24)

Here, some modifications are applied to the derivation presented in [Mei16] so that theexact solution (5.24) holds for contact between beam and a rigid cylinder. The derivation ispresented in Appendix B.

An initial gap distance g = −0.1 · R is considered between two bodies. Th movement ofbeam’s end centerpoint on the circular path in y and z− directions is prescribed as:

∆v = −r

[

1− cos

(k · 2π

N

)]

∆w = r

[

sin

(k · 2π

N

)]

, k = 1, 2, 3, . . . , N

(5.25)

Furthermore, the tangent in y−direction at the beam’s end should be prescribed to zero(∆tv = 0).

Finally, the problem setting completes with definition of contact penalty parameter whichdirectly follows from derivation as

ǫ = −EIh2r

(h2 + r2)3 g0. (5.26)

The problem is solved using a displacement control solver (Sec.2.3) with prescribed boundaryconditions (5.25) and N = 100 load steps. Fig.5.5 shows the relative L2−error for discretiza-tions with 1, 2, 4, 8, 16, 32 elements. The numerical convergence order is approximately 3.5which is slightly lower than 4thorder convergence rate of the employed beam element formula-tion. The difference could be due to the contact element contribution which is not consideredin L2−error convergence rate (4.52). Nevertheless, this example shows convergence behaviorof the presented contact formulation.


10−1 100 10110−12

10−9

10−6

10−3

100

h(element length)

‖e‖2 r

el

O(h4)numerical solution

O(h3.58)

Figure 5.5: Beam Twist Benchmark

55

Chapter 6

Modeling Interactions of the

Wiring and Robot Linkages

In this chapter, the developed beam and contact elements are used to incorporate flexiblewiring deformations into kinematic simulation of an industrial robot. This kind of robots canbe viewed as a set of joined links with the ability to move each link separately in rotational ortransnational manner. When modeling deformations of wires attached to the links, movementof the robot linkages are necessary to be defined, since it in turn specifies the movement ofattached wires. In addition to this, in order to model contact interactions between wires andthe robot links, geometry of the links and their position in space at each solution time stepshould be known. In this work, the robot links are considered as analytic surfaces, namelycylinders and spheres. To define the links’ positions, robot kinematics is discussed in Sec.6.1.The definition of wiring boundary conditions is subject of Sec.6.2.1. In the final section ofthis chapter, Sec.6.2.3, discussed concepts are employed in two numerical examples.

6.1 Robot Kinematics

Kinematics of robots treats the movement of links without considering forces and torquesthat cause the motion. In other words, using kinematics it is possible to define the positionand orientation of the robot links. Movements of the links are defined by joints. In design ofindustrial robots, only one degree of freedom is considered for each joint, either rotation ortranslation. The former is called a revolute joint while the latter is a prismatic joint. In thissection, robots with one DOF per joint are treated. Two neighboring joints are connectedwith one link. The most commonly used industrial robot consists of 6 revolute joints and iscalled a 6R manipulator (Fig.6.1).

While robot links in the engineering sense consist of a set of geometrical and mechanicalparameters, in the kinematic sense, robot links are rigid bodies that connect two neighboringjoints. If robot joints are numbered in successive order, link i is positioned among jointsi and i + 1(Fig. 6.2). A mathematical tool to refrain from dealing with the complexshapes of robot links would be affixing frames to each link and define movement of thelinks with help of transformation matrices. A widely used notation to define these framesis Denavit-Hartenberg (DH) notation. In the following, first the DH parameters and link

56 6. Modeling Interactions of the Wiring and Robot Linkages

Figure 6.1: A simple drawing of a 6R manipulator with analytic surfaces and rotation axes

joint i joint i+ 1

link i

Figure 6.2: ith robot link connects joints i and i+ 1

frames are presented. Afterwards, transformation matrices are calculated for each link. Formore detailed information on robot kinematics interested reader is referred to [Cra05].

6.1.1 Denavit-Hartenberg Notation

To define the relative location of link joints in 3D-space, Denavit and Hartenberg used 4parameters for each link in a robot [JR55]. These so-called DH parameters for link i are:link length ai, link offset di, joint angle θi, and link twist αi. To specify these parametersand attached frames to each link, one should follow a predefined instruction.

First, all joint axes of the robot should be defined and successively numbered. For revolutejoints, this is the axis around which robot link rotates and for prismatic joints, the robot linkmoves along the joint axis. Two neighboring lines i and i + 1 define the ithlink. Next,the common normal to these two lines is identified. The ithLink-frame’s origin is assignedat intersection point of the common normal and the joint axis i. Also, the Zi axis pointsalong this joint axis and the Xi axis points along the common perpendicular and in directionfrom axis i to axis i+1. If two lines intersect, there are two choices to define direction of

6.1. Robot Kinematics 57

a1

x0, x1

y0, y1

z0, z1 x2

z2

y2

d2

x3

z3

y3

a2

d3

a3

d4x4, x5, x6

z4, z6

z5

Figure 6.3: Coordinate frames and DH-parameters of 6R robot in Fig. 6.1

Xi. Finally, axis Yi is assigned such that a right-hand coordinate system is completed. Thisframe is rigidly attached to its corresponding link.

Once the frames are sketched, finding DH parameters is straightforward. We consider framesi− 1, i, and i+1 respectively affixed to links i− 1, i, and i+1. The link lengthof axis i, ai, is the distance between axes Zi and Zi+1 measured along Xi. Similarly, theangle between two axes Zi and Zi+1 along Xi is twist angle of axis i. The two remainingparameters, the link offset di and the joint angle θi, are the distance and the angle from axisXi−1 to axis Xi with respect to Zi. For revolute joints, θi is joint variable and the other threequantities are the link parameters that are fixed, whereas, for prismatic joints di is the jointvariable.

Using the above-mentioned instructions, intermediate frames and DH parameters in a kine-matic chain with n links can be specified. To attach the last frame of the chain, frame n,we define Xn in such that it aligns with frame Xn−1 when the joint variable, θ for revoluteand d for prismatic joint, is zero. Furthermore, we define a fixed reference frame, frame 0.Since this frame is arbitrary, for the sake of simplification, it is preferable to choose this frameto be aligned with frame i when the joint variable of frame one is zero.

Fig.6.3 and Table 6.1 illustrate DH joint parameters and link frames of robot in Fig.6.1.

6.1.2 Robot Transformation Matrices

In the last section, frames affixed to robot links and DH parameters, with which kinematicsproperties of the links can be fully defined, are handled. Now, we treat the calculation of the


Table 6.1: DH-parameters of 6R robot in Fig.6.1

d a α θ

0 – a0 = 0 α0 = 0 –1 d1 = 0 a1 α1 =

π2 θ1

2 d2 a2 α2 = π θ23 d3 a3 α3 = −π

2 θ34 d4 a4 = 0 α4 =

π2 θ4

5 d5 = 0 a5 = 0 α5 = −π2 θ5

6 d6 = 0 – – θ6

position and the orientation of the nth link relative to the fixed reference frame 0. To thisend, a transformation matrix should be constructed that relates the coordinates in frame n tothe coordinates in frame 0, which is denoted by 0

nT . The transformation matrix 0nT is broken

into sub matrices i−1i T , which defines coordinates of link i relative to i− 1. Thereupon,

these individual matrices are multiplied successively in order to form the overall matrix 0nT .

Link Transformation Matrix

The transformation matrix i−1i T , with which coordinates in frame i, iP , can be calculated

with respect to coordinates in frame i − 1, i−1P , is a function of the four DH parameterscorresponding to link i. As shown in [Cra05], construction of this matrix results from foursuccessive rotations and translations as

i−1P =[i−1R T

] [RQT

] [QPT

] [Pi T

] [iP

], (6.1)

where,

• i−1R T is rotation around axis Xi−1 by angle αi−1

• RQT is translation along axis Xi−1 by distance ai−1

• QPT is rotation around axis Zi by angle θi

• Pi T is translation along axis Zi by distance di

Considering each of these transformation, general form of i−1i T is obtained:

i−1i T =

cθi −sθi 0 ai−1

sθicαi−1 cθicαi−1 −sαi−1 −sαi−1disθisαi−1 cθisαi−1 cαi−1 cαi−1di

0 0 0 1

(6.2)

6.2. Robot and Wire Interactions 59

Here, cos and sin are denoted by c and s. To use the transformation matrix (6.2), thecoordinates iP and i−1P should have the following format

iP =

xiyizi1

(6.3)

Here, unit entry in 4th row corresponds with the translations along axis Xi−1 and Zi.

Overall Transformation Matrix

In the last step, transformation matrices of individual links are calculated. Transformationmatrix from frame n to frame i, i

nT can be computed by successively multiplying theindividual transformations i to n as

inT =

[ii+1T

] [i+1i+2T

]...[n−1

n T ]. (6.4)

Hence, the Cartesian position of an arbitrary link n in reference frame 0 is calculated via thetransformation matrix 0

nT . This matrix is a function of n joint variables.

6.2 Robot and Wire Interactions

In addition to links and their mechanisms, a robot consists of motors and sensors. Necessarypower for the motors, and data read by the sensors are transformed through robot wires.In many designs, wires are packed in a wire protection cover and externally attached tothe links. During movement of a robot, especially in 6R type with their high capabilityof complex motions, there are configurations where the attached wires might get damagedand hence hinder robot maneuverability. The availability of a tool to model deformations ofexternal wiring can help engineers to better plan robot movements in order to prevent thesedamages. Also, it can be used to come up with better robot linkage designs and mounting ofcables.

In this section, the kinematic simulation of industrial robots is extended with deformationof wires attached to them. These deformations are determined by two major effects: themovement of the attached wiring sections to the robot, and contact interactions betweenwiring and robot links. In some cases, depending on the wiring stiffness, its weight mightalso play a role.

To compute the deformations, the robot movement during the kinematic simulation is dividedinto time steps with size ∆t. For more complex parts of motion, a finer ∆t might be chosen.At each time step, the displacement of wiring attached points to robot is calculated. Thisinformation provides boundary conditions for the numerical simulation of the deformations.Cables in the simulation are assumed as beams with a thin cross section (negligible shearstresses). Deformations are computed at each time step statically without considering inertiaeffects. Following from this, our simulation can be regarded as quasi-static. Some wire


Figure 6.4: Coupling between robot kinematics and cable simulator modules

attachment systems also prohibit the rotation of cables at the attachment points. This canbe realized by prescribing the direction of boundary nodes in the beam. Calculation ofpoints’ coordinates and their tangent vectors is treated in Sec.6.2.1. Furthermore, in contactelements, beams are considered as slave surfaces and robot’s rigid links as the master side.At each time step, if the robot links move, their position and orientation should be updated.This is done with the kinematic parameters of the robot and the transformation matricesdefined in the last section.

Nodal displacements of beam elements with prescribed boundary movements are solved usingthe nonlinear Newton-Raphson method with displacement control path following method (seeSec. 2.3). Solution steps are identical with movement time steps. The expression “solutiontime step” is used in the following to refer to the both. Fig. 6.4 illustrates how robotkinematic and cable simulator module are coupled in one solution time step.

Modeling of weight effects is one more topic that is treated in this section. In industrial robotdesigns, usually a group of wires and cables are packed together inside a cable protection layer.In this case, the cable protection is stiff enough such that the weight could be neglected.However, in other cases where the wiring system is more deformable, or a single cable issubject of simulation, the weight should be considered. It is to be noted that the interactionbetween cables is not the focus of this work. In case of weight effects, the displacementcontrol solver should be extended with external loading. Sec. (6.2.2) discusses this and allother necessary adaptations to the solver including limiting displacement increments so as tohinder the beam elements jumping over contact targets, and correct physical positioning ofthe beam nodes in an initial configuration.


pi

ti

xi xi+1

zi+1

yi+1zi

yi

x0

z0

y0

Figure 6.5: Wire attached to link i at point p

6.2.1 Calculation of Cable Boundary Conditions

In this section, the position and tangent vectors of wiring attachment points to robot arecalculated. Depending on the problem at hand, just the position or both (position andtangent) might be necessary as boundary conditions. We assume the wire is affixed to robotlink i as shown in Fig.6.5. Attachment position and its arbitrary tangent vector at robotframe i are iP = (Pxi

, Pyi , Pzi) andit = (txi

, tyi , tzi), respectively.

Additionally, the wiring material frame is Ew1 ,E

w2 ,E

w3 . The robot reference frame, frame

0 as discussed in Sec.6.1.2, is denoted by E0,E1,E2. To simplify calculations, we assumeboth frames coincide. Position and tangent vectors should be defined in the wiring materialframe.

The position of attachment point, iP , at frame i is transformed to the reference frameusing the transformation matrix i

0T , defined by (6.4) as

Px0

Py0

Pz0

1

=[0i T

]

Pxi

Pyi

Pzi

1

. (6.5)

The same transformation matrix can be used for it. However, since a vector is transformed,only the rotational part of i

0T is necessary. To cancel out the translation part, fourth com-ponent of (6.3) is substituted with 0.

tx0

ty0tz00

=[0iT

]

txi

tyitzi0

(6.6)


A common direction for the fixed attachment point in robots is parallel to link. In this case,the tangent vector it is parallel to axis zi in frame i

txi

tyitzi

=

001

(6.7)

Remark 6. As discussed in Remark 3, to model Drichlet boundary conditions in the GE beamformulation, the magnitude of tangent vector should not be fixed. This was done by leaving outthe component of the tangent vector that is parallel to the beam initial longitudinal axis. Thesame holds here. In other words, if the cable initially lays along Ew

i , then the correspondingcomponent of the tangent vector should not be initialized.

6.2.2 Adaptations to Displacement Control Solver

For modeling wire deformations, due to the prescribed position of wire attachment points,in general a Newton-Raphson nonlinear solver with displacement path following method isutilized. However, some tweaks to the method described in Sec. 2.3 are necessary to make itapplicable to this special application. This section treats such adaptations.

Extending Displacement Control Solver with External Loading

In cases where the weight effect is to be considered in wire deformations, a constant lineload should be applied to the beam elements. Hence, both prescribed displacements andexternal load are present at the solution. In Sec. 2.3, the introduced displacement controlsolver can account for external loads. However, it is not possible to apply the load in anincremental manner. This might be problematic for solution convergence if the load valueis high. One possible method to deal with such case is: first, the wire deformations due toload are computed using a load control path following technique. Afterwards, the solution iscontinued using displacement control method to calculate deformations due to movement ofattachment points. Considering path-independence of the employed beam element, this doesnot affect the final position of the beam centerline. Also, from a physical point of view, thisresembles the fact that deformations due to weight are present in the system before movementof the robot links.

Displacement Increments Control

Another problem that might occur during the simulation of wire deformations is the unde-tected crossing of wires and robot links without contact being detected. If the displacementincrement norm of beam i at solution iteration k, ‖∆dK

i ‖ is larger than the diameter of jth

robot link, 2Rj , then the mentioned beam can cross the link with gap function g still beingpositive and hence no contact being detected.

To prohibit this scenario, one trivial solution would be setting solution step size to smallvalues. However, such an approach involves either choosing a very small times step size,which deteriorates numerical solution efficiency, or a number of try-and-errors before a rather


optimal choice of step size, which can prohibit undetected contact crossings and requires theleast possible time steps at the same time.

An automatic alternative approach is to confine the element displacement increments tomaximal permissible displacement increment dmax. For the following, we assume ∆Dk

mid

being displacement increment of beam elements midpoints at arbitrary iteration k,

∆Dkmid = ∆dk

1,mid,∆dk2,mid,∆dk3,mid..., dkn,mid (6.8)

At each iteration inf-norm of ∆Dkmid, ‖∆Dk

mid‖∞, is computed. If it is higher than dmax, the

whole nodal displacement increment vector ∆dk is scaled by a factor to limit max the maxincrement. The proposed approach can be formulated as

if ‖∆Dkmid‖∞ > dmax then ∆dk

mod ≡dmax

‖∆Dkmid‖∞

∆dk, (6.9)

where, ∆dkmod defines modified nodal displacement increment vector, and ‖∆Dk

mid‖∞ is de-fined as

‖∆Dkmid‖∞ ≡ max‖∆dk

1,mid‖, ‖∆dk2,mid‖, ‖∆dk3,mid‖, ..., ‖dkn,mid‖. (6.10)

Wiring Initialization

One more tweak that should be considered in the solution process is the initialization of therobot’s wires. While Bezier or B-spline curves can be used to model one-dimensional curvesin 3D space, this might not resemble the natural initial shape of the wires. To deal withthis case, an initialization step is defined in the solution process. Starting with discretizedinitially straight beam, the natural form is found using displacement control with prescribedposition and tangent (if desired) vectors of wire attachment point to robot links. Afterwards,simulation of wire deformations due to attachment points movement can be continued.

6.2.3 Numerical Examples

The previously discussed torsion-free geometrically exact beam element of Kirchhoff-type(chapter 4), Gauss point to analytic surface normal contact (chapter 5), and practical issues inthe current chapter are put together in order to simulate wire deformations in robotic arms inthis section. In the following, two numerical examples are subject to study. First, in a contactdominant motion configuration, the wire deformations are modeled. In the second example,a real-world robot motion is simulated. Performance of the employed Newton-Raphson solverwith displacement control path following technique in terms of average number of iterationsper load step and numerical efficiency in terms of total solution time is measured. Thesimulations are conducted on a machine with Intel Core i7-2670 2.2 GHz CPU and installedRAM of 6.0 GB.

In both examples, the physical and geometrical properties of the wiring are: circular crosssection with radius r = 20mm and Young’s modulus E = 1.0 × 107Pa. The weight of thewiring is also considered by applying a constant line load of w = 57N/m.


f‖

m0

1

2

Figure 6.6: Stress resultants at beam cross section

Deformation results at the wiring are represented either via contact forces along the beamcenterline or stresses at its cross section. Contact forces can simply be calculated as

fc = −ǫ〈g(ξ)〉 (6.11)

The same notation as Sec. 5.1 is used here. Having element i nodal displacement vector di,gap distance at desired parametric coordinate ξ ∈ (−1,+1) is calculated by (5.4). If its valueis negative, the contact force fc is nonzero. Similarly, using di stress resultants f‖ and m atthe beam centerline are computed by relations discussed in Sec. 4.2.2. Considering Fig. 6.6and using stress relations from general beam theory, stresses at points 0, 1, 2 are

σ0 = f‖ (6.12a)

σ1 = −m ·R

I+ f‖ (6.12b)

σ2 = +m · R

I+ f‖. (6.12c)

Here, R and I are the wire radius and area moment of inertia.

Example 1: Wiring Twist around Robot Link

In this example, twisting of the wiring around robot link is modeled. Affixed frames tothe robot links to describe its kinematics is similar to Fig.6.3 with the specification of DH-parameters presented in Table 6.2.

Wiring with initial length L0 = 4m is attached to the robot at two points. The wire’s begin-ning point is attached to the robot at the second link with coordinates 2P = (1.5, 0.0, 0.294)T

and no prescribed direction of tangent vector. Also, the wire’s endpoint is connected to the4th link with coordinates 4P = (0.0, 0.244,−1.0)T and parallel tangent vector with the link


Table 6.2: Example1: Specification of DH parameters

d[m] a[m] α[rad] θ[rad]

0 – a0 = 0 α0 = 0 –1 d1 = 0 0.9 α1 =

π2 θ1

2 0.675 3.0 α2 = π θ23 0.675 0.6 α3 = −π

2 θ34 2.0 a4 = 0 α4 =

π2 θ4

5 d5 = 0 a5 = 0 α5 = −π2 θ5

6 d6 = 0 – – θ6

Table 6.3: Example 2: Wiring attachment points and BCs

point No. link No. Coordinates [mm] tangent

1 2 2P = (1050.0, 0.0, 82.2)T 2t = (−1.0, 0.0, 0.0)T

2 3 3P = (41.0,−300.0, 50.0)T 3t = (−1.0, 0.0, 0.0)T

3 6 6P = (30.0, 27.2,−80.0)T 6t = (0.0,−12 ,−

√32 )T

4t = (0.0, 0.0, 1.0)T . The mentioned link is rotated 2π radians around its rotation axis duringkinematic simulation of motion. As a result, the wiring is expected to twist around it dueto rather heavy contact interactions that happen during the motion. The contact penaltyfactor is set to ǫ = 1 · 106. Furthermore, the wiring is discretized by 16 beam elements withequal length.

The robot motion is subdivided into 100 solution time steps with equal timing. Additionally,20 time steps are considered at the beginning of the simulation to find initial wiring shapeas well as to apply its weight.

Fig.6.7 represents state of the deformation during the motion. The total simulation includingthe initialization steps takes on average 11.5 seconds on the machine. Total number of solveriterations for robot motion time steps (exclusive of is initialization steps) is 956, which meansan average of 9.5 iterations per solution time step.

Example 2: Real World Robot Motion

Deformation of the wiring while a 6R industrial robot maneuvers with a real-world motion ismodeled. The purpose of this example is to demonstrate the applicability of the introducedconcepts to practical examples. A Kuka industrial robot, model Kr120-r2700 is used to thisend. The kinematic values such as link length, twist, etc. are the correct measure ones. Fig.6.8 illustrates the reproduced geometric model of the robot using analytic surfaces (spheresand cylinders). Kinematic parameters of this model are measured from Kr120-r2700, however,other geometric values e.g. radii of cylinders and spheres are set such that they fit to theother dimensions of the model.

Wiring with initial length L0 = 4 is discirtized with 25 beam elements. Attachment pointsto the links and BCs are shown in Table. 6.3.

Furthermore, the contact penalty factor used in contact elements is set to ǫ = 1 · 106.


(a)

(b) (c)

(d)

(f)

(e)

(g)

Figure 6.7: Example 1: Wiring deformation state at different frames during the motion, a) Wiringinitial configuration, b) t = 0 (wiring initial physical shape), c) t = 20, d)t = 40, e) t = 60, f) t = 80,g) t = 100


Figure 6.8: Reproduced analytic model from Kuka Kr120-r2700 robot as contact analytic surface

Table 6.4: Example 2: Joint variables at time steps with more complex deformation state of thewiring

time step θ1 θ2 θ3 θ4 θ5 θ60 0. -120. 140. 0. -20. 0.39 -75.3 -55.7 109.3 14.2 -55.7 -8.261 -75.2 -60.1 109.4 15.1 -51.5 -9.6234 -87.8 -50.8 83.6 44.4 27.9 -25.1308 -81.4 -53.6 86.5 32.7 33.1 -79.4433 -90.2 -51.6 87.5 -132.2 -76.6 5.5

The total robot motion time is 37 seconds which is divided into 500 solution time steps.General method to choose the number of time step is the more solution time steps areassigned to sub-domains where robot end effector moves faster, and hence, unlike the lastexample, solution steps do not have equal timing. Additional 40 time steps are used forwiring initialization and applying its weight.

Table 6.4 summarizes the robot’s joint variables at time steps at which the wiring deformsin a rather more complex manner. The resulted deformation states are presented in fig. 6.9.Total simulation time take an average of 83.3 seconds, with 4975 solution iteration during400 time steps of motion (average of 9.9 per time step).


(a) (b)

(c) (d)

(e) (f)

Figure 6.9: Example 2: Wiring deformation state at different time steps during the motion, a)Wiring initial configuration (t = 0), b) t = 39 (wiring initial physical shape), c) t = 61, d)t = 234, e)t = 308, f) t = 433

69

Chapter 7

Summary and Outlook

Summary

In this work, the kinematic simulation of industrial robots is extended with the elastic defor-mation of the wiring mounted on their links, in order to define critical stresses and deforma-tion states during robots’ motion. To this end, the wiring is modeled by beam elements andfrictionless contact interactions of the wiring and the robot linkages are considered.

A nonlinear beam element formulation is employed to model the wiring since it undergoeslarge deformations. Additionally, due to low slenderness ratio and negligible torsional stiffnessin cables, shear- and torsion-free assumptions are made. In a recent work, Meier [Mei16]developed a geometrically exact beam element formulation of Kirchhoff type based on Simo-Reissner beam theory. He also presented a simplified version that neglects torsion effectsfor torsion-free applications. Here, this formulation is used. Considering the fact that thevariational index of the weak form of the Simo-Reissner theory is two, and since third orderHerimitian shape functions are used to discretize the weak formulation, convergence orderfour of the L2−error for h−refinement is expected. This is verified by a well-known numericaltest case for nonlinear beam elements for which a closed-form solution is available. In additionto this, in order to investigate the membrane locking effects, this test case is repeated fordifferent beam slenderness ratios. Although for high slenderness ratios the formulation sufferslocking, for typical ratios in cables, it seems that the locking effect is negligible and thereforeno anti-locking algorithm is necessary. In a second test case, the beam is subjected to acomplex 3D loading to investigate the behavior of the employed nonlinear solver. To check thesolution accuracy, the results are compared against the values from an overkill discretizationof a Simo-Reissnear shear-deformable beam element. Although the loading and deformationstate are complex in this case, the nonlinear solver converges to the final solution within anappropriate number of iterations. In terms of solution accuracy, for a reasonable number ofelements, the error is less than one percent which is acceptable in engineering applications.

Normal contact interactions between the wiring and the robot linkages are considered usingthe penalty potential function along the beam element. The resulting weak form of thementioned potential is discretized by Gauss Point to Analytic Surface approach in which robotlinks are described by analytic geometric surfaces such as planes, spheres, and cylinders. Sincethere is a few numbers of such surfaces in the simulation, the global search phase, the firststage in contact detection algorithms, is omitted. Additionally, in the linearization process,

70 7. Summary and Outlook

the linearization of the gap function variation form, a quite computationally demanding task,is ignored at the expense of a few more iterations during the solution. Due to discontinuitiesin the gap function, the contact contribution to the overall tangent stiffness matrix and theresidual forces vector cannot be exactly integrated using the Gaussian quadrature rule. Inan equivalent version to the contact patch test for beams, we showed that two Gauss pointsare enough to integrate the gap function exactly for constant loads. However, for generalcases, in which the loading is probably more complex than constant load, the same numberof integration points used in the beam element formulation is utilized. The developed contactalgorithm shows good convergence behavior for h−refinement in a numerical example withavailable analytic solution.

Finally, to merge the wiring simulator into the kinematic simulator, the coupling between thetwo is calculated. This includes wiring boundary conditions, which are position and tangentvectors of the attachment points, and position of the linkages for contact interactions. Therobot forward kinematics is computed via attached coordinate frames to the robot links andtransformation matrices that define the coordinates in one frame with respect to a referenceframe. Due to prescribed displacement of the attachment points during the robot motion,displacement control path following technique is used to solve for the wiring deformations.To prohibit the beam elements from jumping over the links in one increment, the magnitudeof increments in solution iterations is limited by the smallest link radius in the robot. Thismethod proves sufficient in the performed numerical simulations. However, in some cases itrequires a try and error process to set the number of load steps.

Conclusion and Outlook

The numerical simulations that are performed in Chap.6 show applicability of the developedmethod to simulate wire deformations during robot motion within adequate computationaltime. However, as discussed earlier, in this work robot links are assumed to be analyticgeometric surfaces, e.g. spheres and cylinders, and hence a simplified model of the robot isobtained. Kinematic values of this model are set based on the robot kinematic parameters.However, geometry of the robot links is not considered in the model. The author suggeststhe following two approaches for this limitation:

• In modern industrial robots, link designs contain a complex geometry with many de-tails. Mostly, such details are not important in contact interactions with the wiring.Thus, as contact targets robot links can be approximated by simpler geometries thatenclose them. Bounding volume of links, a widely used concept in the computationalgeometry, might be employed for the approximation. Although this assumption mightaffect the accuracy of results, for engineering applications this is a sensible assumption.Additionally, this approach benefits the numerical efficiency resulting from few existingcontact objects.

• The exact shape of links can also be used for calculation of the contact interactions. Tothis end, each link can be discretized by triangular meshes. To detect contact, closestpoint projection to triangles (Appendix A) should be employed. Also, the global searchphase in the contact detection algorithm might be beneficial in order to have a betterthe computational efficiency.

71

Another factor which is not considered in the deformation of the wiring in this work is inertiaeffects. In some configurations, such as fast movement of robot links or high weight of thewiring, this might be desirable to be considered. To do so, the employed beam formulationshould be extended by a temporal discretization method. In [MPW17], a recently proposedextension of the α-method [CH93] is applied to the GE beam formulation used in this work.

72 7. Summary and Outlook

73

Appendix A

Closest Point Projection to

Analytic Surfaces

Here, closed form solutions for closest point projection xM to some analytic surfaces includingplanes, cylinders, spheres, and linear triangles as well as their normal vector nM at theprojection point are presented.

A.0.1 Projection onto Plane

To find projection of point xs onto a plane which is defined by point x1M on the plane and

unit vector n normal to the plane, we characterize displacement vector v as,

v = xs − x1M , (A.1)

where its parallel to n and perpendicular components are,

v‖ = (v · n)n (A.2a)

v⊥ = v − v‖ . (A.2b)

Coordinates of the projected point are resulted by summation of bmx1M with perpendicularcomponent of v as,

xM = x1M + v⊥. (A.3)

And normal vector nM coincides with vector n.

A.0.2 Projection onto Sphere

We assume a sphere with its center and radius to be xc and r respectively. To define projectionof point xs onto the sphere, displacement vector v from xc to xs is,

v = xs − xc. (A.4)

74 A. Closest Point Projection to Analytic Surfaces

Normalization of v results in the normal vector nM as,

nM =v

‖v‖. (A.5)

Coordinates of the projected point are calculated by moving the sphere center xc along thenormal vector nM with magnitude r:

xM = xc + rnM (A.6)

A.0.3 Projection onto Cylinder

We assume a cylinder with center points at two ends x1c , x

2c and radius r. Also, point xs is

to be projected on lateral surface of the cylinder. To do so, displacement vectors v and vsare characterized as,

v = x2c − x1

c (A.7a)

vs = xs − x1c . (A.7b)

Projection of the point xs onto the cylinder centerline is,

xc = x1c +

v1 · v2‖v1‖2

v1. (A.8)

Unit normal vector at the projection point can be defined as,

nM =xs − xc

‖xs − xc‖. (A.9)

Finally, the projected point coordinates xM is calculated as,

xM = xc + nMr (A.10)

A.0.4 Projection onto Triangle

We consider a linear triangle with corners x1M , x2

M , and x3M . Point xs is to be projected

on the triangle according to method proposed by Heidrich [Hei05]. Hence, we define thefollowing displacement and normal vectors:

v1 = x2M − x1

M (A.11a)

v2 = x3M − x1

M (A.11b)

vs = xs − x1M (A.11c)

n = v1 × v2 (A.11d)

75

Unit normal vector nM can easily be evaluated as,

nM =n

‖n‖. (A.12)

The local barycentric coordinates τM (see e.g. [Bog17]) of the projected point are,

τ2M =(v1 × vs)× n

n · n(A.13a)

τ3M =(v2 × vs)× n

n · n(A.13b)

τ1M = 1.0− τ2M − τ3M . (A.13c)

Transformation of the projected point τM to the global coordinates results in xM as,

xM =

3∑

i=1

xiM · τ iM . (A.14)

76 A. Closest Point Projection to Analytic Surfaces

77

Appendix B

Analytic Solution for Beam Contact

with a Rigid Cylinder

In this appendix, a closed form solution for contact of a beam against a rigid cylinder basedon the work of Meier [Mei16] is derived. While the mentioned work is originally derived forthe case of twisting of two beams and nonzero prescribed axial tension in the beams, herethe case is simplified for twist around a rigid cylinder and zero axial tension. Before treatingthe main problem, first a tool should be introduced which is beneficial for analytic treatmentof the strong form of the GE Kirchhoff beam (4.15).

Analytical Treatment of the GE Kirchhoff Beam

In [MPW15], the Frenet-Serret(FS) reference triad is applied to the strong form (4.15). Asargued in the work, while this triad is not suitable for numerical treatment of the GE Kirchhoffbeam, it makes analytic treatment of the formulation much easier. According to the triad,at each point of the curve, the FS normal vector n points in the direction of the centerof its curvature. Together with the tangent vector t and FS binormal vector, which is thevector product of t and n, the right-handed orthonormal triad is completed. Accordingly, themoment stress resultant m and the external moment and force vectors (m and f respectively)can be written in the FS frame by their components as,

m = m‖t+mnn+mbb (B.1a)

m = m‖t+ mnn+ mbb (B.1b)

f = f‖t+ fnn+ fbb. (B.1c)

Introducing the above relations into the vector valued relations (4.15) results in four differ-ential equations as,

f ′‖ +

κ

1 + ǫ(τmn +m′

b + mb) + f‖ = 0 (B.2a)

78 B. Analytic Solution for Beam Contact with a Rigid Cylinder

−

(τmn +m′

b + mb

1 + ǫ

)′−

τ

1 + ǫ(κm‖ +m′

n − τmb + mn) + κf‖ + fn = 0 (B.2b)

(−τmb +m′

n + κm‖ + mn

1 + ǫ

)

−τ

1 + ǫ(τmn +m′

b + mb) + fb = 0 (B.2c)

m′‖ − κmn + m‖ = 0, (B.2d)

where, similar to the notation uses so far, ǫ, τ , and κ denote axial strain, torsion, and FScurvature respectively. Furthermore, components of the stress resultant f‖, m‖, mn, and mb

for the case of quasi-circular cross-sections can be written as,

f‖ = EAǫ (B.3a)

m‖ = GIT (τ + φ′ − τ0 + φ′0) (B.3b)

mn = EIκ0sinφ (B.3c)

mb = EI(κ − κ0cosφ). (B.3d)

It is worth to mention that the above relations are written for the torsion-deformable beamwith the general initially straight configuration.

Closed Form Solution for Twist around a Rigid Cylinder

The mechanical curvature and torsion of the analytic helix shown by (5.24) per definition is

κ =r(1 + ǫ)

h2 + r2(B.4a)

τ =h(1 + ǫ)

h2 + r2. (B.4b)

where, as defined earlier, h, r, and ǫ denote helix slope, helix radius and axial strain ofthe beam. Also, the constant mechanical curvature of the helix results the derivative of themoment stress resultant in binormal direction to be zero, m′

b = 0.

In static equilibrium, apart from discrete point forces and moments at right and left endsof the beam , which are due to the applied Drichlet conditions, and the line load in theFS normal direction fn, which is due to contact interactions with the rigid cylinder, all theremaining external distributed force and moment components are zero. With the constantgap distance g0 defined in Sec. 5.4.2 and penalty factor ǫc, the normal force component fncan be calculated as,

fn = ǫcg0. (B.5)

In this derivation, we consider a solution with a constant prescribed axial strain ǫ = constand torsion τ + φ′ = const along the beam. These requirement and relations (B.3) results inf ′‖ = 0 and m′

‖ = 0.

Next, we put relations (B.3) with the concluded values so far in the differential equations(B.2). This results in only one remaining nonzero equation that should be satisfied by thesystem parameters. The equation reads:

79

−τ

1 + ǫ(κm‖ − τmb) + κf‖ + fn = 0. (B.6)

Furthermore, we assume a solution with vanishing torsion, m‖ = 0, and the prescribed axialstrain to be zero. Hence, the above equation simplifies as,

h2

(h2 + r2)2EIr

h2 + r2+ ǫcg0 = 0. (B.7)

This postulates a constraint for the contact penalty factor which reads,

ǫc = −EIh2r

(h2 + r2)3g0. (B.8)

With the above value for ǫc all equations in (B.2) are satisfied and therefore the desiredsolution is obtained.

80 B. Analytic Solution for Beam Contact with a Rigid Cylinder

LIST OF FIGURES 81

List of Figures

2.1 General motion of body in 3D space . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Path following methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Contact regularization methods . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1 Kinematics of GE beam theory . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Beam element example 1, pure bending in 2D . . . . . . . . . . . . . . . . . . 42

4.4 Beam element example 1, L2−error . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Beam element example 2, 3D Line loading . . . . . . . . . . . . . . . . . . . . 43

4.6 Beam element example 2, error in middle point displacement . . . . . . . . . 44

5.1 Closest Point Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Contact example 1, patch test . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3 Contact example 1, error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.4 Contact example 2, twist around rigid cylinder . . . . . . . . . . . . . . . . . 52

5.5 Contact example 2, beam twist benchmark . . . . . . . . . . . . . . . . . . . 54

6.1 Simple illustration of 6R manipulator . . . . . . . . . . . . . . . . . . . . . . 56

6.2 ith robot link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.3 Coordinate frames and DH-parameters of 6R robot in Fig. 6.1 . . . . . . . . 57

6.4 Coupling between robot kinematics and cable simulator modules . . . . . . . 60

6.5 Wire attached to link i at point p . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.6 Stress resultants at beam cross section . . . . . . . . . . . . . . . . . . . . . . 64

6.7 Robot example 1: deformation states . . . . . . . . . . . . . . . . . . . . . . . 66

82 LIST OF FIGURES

6.8 Robot example 2: simple model . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.9 Robot example 2: deformation states . . . . . . . . . . . . . . . . . . . . . . . 68

LIST OF TABLES 83

List of Tables

6.1 DH-parameters of 6R robot in Fig.6.1 . . . . . . . . . . . . . . . . . . . . . . 58

6.2 Specification of DH parameters in robot example 1 . . . . . . . . . . . . . . . 65

6.3 Wiring attachment points and BCs in robot example 2 . . . . . . . . . . . . . 65

6.4 Joint variables in robot example 2 . . . . . . . . . . . . . . . . . . . . . . . . 67

84 LIST OF TABLES

BIBLIOGRAPHY 85

Bibliography

[ACJ99] M. A. Crisfield and G. Jelenic. Objectivity of strain measures in geometricallyexact 3D beam theory and its finite element implementation. Proceedings ofthe Royal Society of London. Series A: Mathematical, Physical and EngineeringSciences., 1999.

[AV12] F. Armero and J. Valverde. Invariant Hermitian finite elements for thin Kirchhoffrods. I: The linear plane case. Computer Methods in Applied Mechanics andEngineering, 2012.

[BB79] K.-J. Bathe and S. Bolourchi. Large displacement analysis of three-dimensionalbeam structures. International Journal for Numerical Methods in Engineering,14(7):961–986, 1979.

[BBP+16] A. M. Bauer, M. Breitenberger, B. Philipp, R. Wuchner, and K. U. Bletzinger.Nonlinear isogeometric spatial Bernoulli beam. Computer Methods in AppliedMechanics and Engineering, 303:101–127, 2016.

[BDNLV11] F. Boyer, G. De Nayer, A. Leroyer, and M. Visonneau. Geometrically ExactKirchhoff Beam Theory: Application to Cable Dynamics. Journal of Computa-tional and Nonlinear Dynamics, 6(4):041004–041004–14, 2011.

[BH73] T. Belytschko and B. J. Hsieh. Nonlinear Transient Finite Element Analysiswith Convected Coordinates. International Journal for Numerical Methods inEngineering, 7(3):255 – 271, 1973.

[Ble16] K. U. Bletzinger. Structural Optimization (Lecture Notes). Technical Universityof Munich, Munich, 2016.

[Bog17] T. Bog. Frictionless Contact Simulation Using the Finite Cell Method. PhDthesis, Technical University of Munich, Munich, 2017.

[BP04] F. Boyer and D. Primault. Finite element of slender beams in finite transfor-mations: A geometrically exact approach. International Journal for NumericalMethods in Engineering, 2004.

[BW08] J. Bonet and R. D. Wood. Nonlinear Continuum Mechanics for Finite ElementAnalysis. Cambridge Univ. Press, Cambridge, second edition, 2008. OCLC:247839016.

[CH93] J. Chung and G. M. Hulbert. A Time Integration Algorithm for StructuralDynamics With Improved Numerical Dissipation: The Generalized-α Method.Journal of Applied Mechanics, 60(2):371–375, 1993.

86 BIBLIOGRAPHY

[CMM09] M. Chamekh, S. Mani-Aouadi, and M. Moakher. Modeling and numerical treat-ment of elastic rods with frictionless self-contact. Computer Methods in AppliedMechanics and Engineering, 198(47):3751–3764, 2009.

[CMM14] M. Chamekh, S. Mani-Aouadi, and M. Moakher. Stability of elastic rods withself-contact. Computer Methods in Applied Mechanics and Engineering, 279:227–246, 2014.

[Cra05] J.J. Craig. Introduction to Robotics: Mechanics and Control. Addison-WesleySeries in Electrical and Computer Engineering: Control Engineering. Pear-son/Prentice Hall, 2005.

[Cri97] M. A. Crisfield. Non-Linear Finite Element Analysis of Solids and Structures:Advanced Topics. John Wiley & Sons, Inc., New York, 1st edition, 1997.

[DBCRV12] R. De Borst, M. A. Crisfield, J. J. C. Remmers, and C. V. Verhoosel. Non-LinearFinite Element Analysis of Solids and Structures. Wiley, second edition, 2012.

[Dur10] D. Durville. Simulation of the mechanical behaviour of woven fabrics at thescale of fibers. International Journal of Material Forming, 3:1241–1251, 2010.

[Dur12] D. Durville. Contact-friction modeling within elastic beam assemblies: An ap-plication to knot tightening. Computational Mechanics, 49(6):687–707, 2012.

[Fel04] C.A. Felippa. Intorduction to Finite Element Methods (Lecture Notes). Univer-sity of Colorado, Colorado, 2004.

[FW05] K.A. Fischer and P. Wriggers. Frictionless 2D Contact formulations for finite de-formations based on the mortar method. Computational Mechanics, 36(3):226–244, 2005.

[FZ75] A. Francavilla and O. C. Zienkiewicz. A note on numerical computation of elasticcontact problems. International Journal for Numerical Methods in Engineering,9(4):913–924, 1975.

[GC13] L. Greco and M. Cuomo. B-Spline interpolation of Kirchhoff-Love space rods.Computer Methods in Applied Mechanics and Engineering, 256:251–269, 2013.

[GC16] L. Greco and M. Cuomo. An isogeometric implicit G1 mixed finite element forKirchhoff space rods. Computer Methods in Applied Mechanics and Engineering,298:325–349, 2016.

[Hei05] W. Heidrich. Computing the Barycentric Coordinates of a Projected Point.Journal of Graphics Tools, 10(3):9–12, 2005.

[HK06] G.J. Hall and E.P. Kapser. Comparison of element technologies for modelingstent expansion. Journal of Biomechanical Engineering, 128(5):751–6, 2006.

[HTK77] T. J. R. Hughes, R. L. Taylor, and W. Kanoknukulchai. A simple and efficientfinite element for plate bending. International Journal for Numerical Methodsin Engineering, 11(10):1529–1543, 1977.

BIBLIOGRAPHY 87

[JC99] G. Jelenic and M.A. Crisfield. Geometrically exact 3D beam theory: Implemen-tation of a strain-invariant finite element for statics and dynamics. ComputerMethods in Applied Mechanics and Engineering, 171(1):141–171, 1999.

[JR55] J.Denavit and R.S. Hartenberg. A Kinematic Notation for Lower-Pair Mecha-nisms Based on Matrices. Journal of Applied Mechanics, pages 215–221, 1955.

[Kir59] G. Kirchhoff. Ueber das Gleichgewicht und die Bewegung eines unendlich dunnenelastischen Stabes. Journal fur die reine und angewandte Mathematik, 56:285–313, 1859.

[Kon11] A. Konyukhov. Geometrically Exact Theory for Contact Interactions. PhDthesis, KIT Scientific Publishing, Karlsruhe, 2011.

[Kon15] A. Konyukhov. Geometrically Exact Theory of Contact Interactions - Applica-tions with Various Methods FEM and FCM. Journal of Applied Mathematicsand Physics, 3(08):1022–1031, 2015.

[Kre09] S. Krenk. Non-Linear Modeling and Analysis of Solids and Structures. Cam-bridge University Press, Cambridge, 2009.

[KS07] A. Konyukhov and K. Schweizerhof. Closest point projection in contact me-chanics: Existence and uniqueness for different types of surfaces. PAMM,7(1):4040053–4040054, 2007.

[LBH14] T.-N. Le, J.-M. Battini, and M. Hjiaj. A consistent 3D corotational beam ele-ment for nonlinear dynamic analysis of flexible structures. Computer Methodsin Applied Mechanics and Engineering, 269:538–565, 2014.

[Lov44] A.E.H. Love. A Treatise on the Mathematical Theory of Elasticity. Dover, NewYork, 1944.

[LS96] J. Langer and D. A. Singer. Lagrangian Aspects of the Kirchhoff Elastic Rod.SIAM Rev, 38(4):605–618, 1996.

[LY08] D. G. Luenberger and Y. Ye. Linear and Nonlinear Programming. Springer,NY, 3rd edition, 2008.

[Mei16] C. Meier. Geometrically Exact Finite Element Formulations for Slender Beamsand Their Contact Interaction. PhD thesis, Technical University of Munich,Munich, 2016.

[MPW14] C. Meier, A. Popp, and W. A. Wall. An objective 3D large deformation finiteelement formulation for geometrically exact curved Kirchhoff rods. ComputerMethods in Applied Mechanics and Engineering, 278:445–478, 2014.

[MPW15] C. Meier, A. Popp, and W. A. Wall. A locking-free finite element formulationand reduced models for geometrically exact Kirchhoff rods. Computer Methodsin Applied Mechanics and Engineering, 290:314–341, 2015.

[MPW16] C. Meier, A. Popp, and W. Wall. A Finite Element Approach for the Line-to-Line Contact Interaction of Thin Beams with Arbitrary Orientation. ComputerMethods in Applied Mechanics and Engineering, 308, 2016.

88 BIBLIOGRAPHY

[MPW17] C. Meier, A. Popp, and W. A. Wall. Geometrically Exact Finite Element Formu-lations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory.Archives of Computational Methods in Engineering, pages 1–81, 2017.

[MWP17] C. Meier, W. A. Wall, and A. Popp. A unified approach for beam-to-beamcontact. Computer Methods in Applied Mechanics and Engineering, 315:972–1010, 2017.

[NMP14] A. G. Neto, C. A. Martins, and P. M. Pimenta. Static analysis of offshorerisers with a geometrically-exact 3D beam model subjected to unilateral contact.Computational Mechanics, 53(1):125–145, 2014.

[NP81] A. K. Noor and J. M. Peters. Mixed models and reduced/selective integra-tion displacement models for nonlinear analysis of curved beams. InternationalJournal for Numerical Methods in Engineering, 17(4):615–631, 1981.

[NPW13] A. G. Neto, P. M. Pimenta, and P. Wriggers. Contact between rolling beamsand flat surfaces. International Journal for Numerical Methods in Engineering,97(9):683–706, 2013.

[PGW09] A. Popp, M. W. Gee, and W. A. Wall. A finite deformation mortar contactformulation using a primal–dual active set strategy. International Journal forNumerical Methods in Engineering, 79(11):1354–1391, 2009.

[Pop17] A. Popp. Computational Contact and Interface Mechanics(Lecture Notes). Tech-nical University of Munich, Munich, 2017.

[Rei72] E. Reissner. On one-dimensional finite-strain beam theory: The plane prob-lem. Zeitschrift fur angewandte Mathematik und Physik (ZAMP), 23(5):795–804,1972.

[Rei81] E. Reissner. Finite deformations of space-curved beams. Zeitschrift fur ange-wandte Mathematik und Physik(ZAMP), 32:734–744, 1981.

[SB82] H. Stolarski and T. Belytschko. Membrane Locking and Reduced Integrationfor Curved Elements. Journal of Applied Mechanics, 1982.

[SF08] G. Strang and G. Fix. An Analysis of the Finite Element Method. Wellesley-Cambridge Press, 2008.

[SH94] Y. Shi and J.E. Hearst. The Kirchhoff elastic rod, the nonlinear Schrodingerequation, and DNA supercoiling. The Journal of Chemical Physics, 101(6):5186–5200, 1994.

[Sim85] J.C. Simo. A finite strain beam formulation. The three-dimensional dynamicproblem. Part I. Computer Methods in Applied Mechanics and Engineering,49(1):55–70, 1985.

[SKIS14] K. Schweizerhof, A. Konyukhov, R. Izi, and M. Strobl. A Solid Beam Elementfor Wire Rope Simulation with A Special Contact Algorithm. 2014.

[SWT85] J. C. Simo, P. Wriggers, and R. L. Taylor. A perturbed Lagrangian formulationfor the finite element solution of contact problems. Computer Methods in AppliedMechanics and Engineering, 50(2):163–180, 1985.

BIBLIOGRAPHY 89

[Tim21] S.P. Timoshenko. On the correction for shear of the differential equationfor transverse vibrations of prismatic bars. Philosophical Magazine Series,41(245):744–746, 1921.

[TP91] R.L. Taylor and P. Papadopoulos. On a patch test for contact problems in twodimensions. In Computational Methods in Nonlinear Mechanics, pages 690–702.Springer, Berlin, 1991.

[WBB16] R. Wuchner, M. Bischoff, and K. U. Bletzinger. Advanced Finite Element Meh-ods (Lecture Notes). Technical University of Munich, Munich, 2016.

[Wei02a] H. Weiss. Dynamics of Geometrically Nonlinear Rods: I. Mechanical Modelsand Equations of Motion. Nonlinear Dynamics, 2002.

[Wei02b] H. Weiss. Dynamics of Geometrically Nonlinear Rods: II. Numerical Methodsand Computational Examples. Nonlinear Dynamics, 2002.

[Wem69] G. Wempner. Finite elements, finite rotations and small strains of flexible shells.International Journal of Solids and Structures, 5(2):117–153, 1969.

[Woh01] B.I. Wohlmuth. Discretization Methods and Iterative Solvers Based on DomainDecomposition. Springer, 2001.

[Wri06] P. Wriggers. Computational Contact Mechanics. Springer, second edition, 2006.

[Wri08] P. Wriggers. Nonlinear Finite Element Methods. Springer Berlin Heidelberg,2008.

[WVVS90] P. Wriggers, T. Vu Van, and E. Stein. Finite element formulation of largedeformation impact-contact problems with friction. Computers & Structures,37(3):319–331, 1990.

[WZ97] P. Wriggers and G. Zavarise. On contact between three-dimensional beamsundergoing large deflections. Communications in Numerical Methods in Engi-neering, 13(6):429–438, 1997.

[Yas11] V. Yastrebov. Computational Contact Mechanics: Geometry, Detection andNumerical Techniques. PhD thesis, Ecole Nationale Superieure des Mines deParis, 2011.

Modeling of Flexible Wirings and Contact …...robot linkages and mounting of the wiring in terms of...

Documents

Transcript of Modeling of Flexible Wirings and Contact …...robot linkages and mounting of the wiring in terms of...