Complex Cable Bundle Simulation and Validation in VR

Christian WienssIC:IDO GmbH, Cologne, Germany

christian.wienss@icido.com

Julia Scharping, Stefan MullerUniversity of Koblenz

Koblenz, Germany

Igor Nikitin, Gernot Goebbels, Martin Gobel, Nils HornungfleXilution GmbHCologne, Germany

Abstract

In this paper a physically accurate real time cable sim-ulation is presented and validated. The simulation methodis fully integrated in Virtual Reality, which increases the us-ability and the sense for physical realism. The influence ofpredeformation, non-homogeneous cross sections and dif-ferent taping of the bundles will be analyzed and comparedto the simulation. The simulation is validated with respectto the physical behavior of real cables and bundles. Ma-terial properties like stiffness and density are measured ofevery specimen. To avoid further expensive measurements,a way to determine bending and torsional stiffness numer-ically with the knowledge of the cross section is presentedand analyzed.

Figure 1. Real time cable simulation in VR[12].

1. Introduction

Since the construction and building of real mock-ups iscostly and time-consuming, the proportion of virtual simu-lation in the product life cycle grows. Today e.g. visibility,reachability and ergonomic questions can be dealt with longbefore the first real mock-up is constructed. The knowledgeabout the accuracy of the utilized simulation tool is indis-pensable for the substitution of mock-ups by virtual proto-typing. The motion and behavior of stiff objects can alreadybe dealt with high performance.

Concerning flexible parts, the usage of simulation is notvery common since the tools usually lack accuracy, usabil-ity or real-time performance. Non-physical representationslike splines dominate today’s digital work with cables andhoses. This provides the user with real-time interaction, butthe physical behavior is disregarded. In order to contrastwith these approaches, a simulation tool has to demonstrateits physical correctness. To prove accuracy, one needs thereal material parameters of a real complex cable bundle andcompare the simulation results with reality.

The focus of previous publications was concentrated onthe real-time solution and collision handling [5], the dy-namic behavior [2], and the mathematical background ofthe approach [8, 9, 5, 2]. In this paper, the accuracy of thesystem is tested and compared with reality. Therefore, realuse case setups and advanced trial setups are taken into ac-count.

2. Related Work

Publications dealing with the simulation of flexible ob-jects can be classified depending on the research progress.The first step is the approach itself with few optimizations[4, 8, 19]. If the idea is promising, the next goal is to ob-

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tain real-time performance for smaller segments and smallexaminations of the accuracy are done [15, 3, 18]. Nowthe simulation is embedded into some virtual environment,where additional functionality, usability and features are ap-plied without great additions to the simulation core [7, 9].After that is successfully done, a product is developed outof this approach. Selling this product means to be con-fronted with the customers’ demands for interaction, inte-gration and their use cases [5, 2]. To replace step by stepphysical mock-ups with virtual simulation, a high credibil-ity of the simulation tool is needed which leads to the nextstep: the proof of concept, which is dealt with in this paper.

Most publications concerning simulation of cables andhoses include sections where the accuracy of the approachis embraced, but none of them prove their correctness ex-pressively.

Theetten [17, 18] presents his work on geometrically ex-act dynamic splines and follows the ideas of [16, 10, 11].Here the comparison has been made between the simula-tion and the theoretical solution of the elliptic equations.Several loads are applied to a 4 meter cantilever beam andthe bending behavior of the presented approach (GEDS) iscomputed. No numbers are given, only a visual comparisonand depictions like ”‘close”’ and ”‘accurate”’ with relationto the number of control points.

The idea of Schotte [15], based on [14, 13, 1], is the mod-ification of a mass-spring system, where forces are replacedby impulses. The accuracy of this iterative model dependsof the value of the distance condition Dmax. If Dmax is setto a large value, the computation is fast, but the accuracy islow and vice versa. Unfortunately Schotte makes no valida-tion of the results and only mentions the accuracy increasesif Dmax is reduced.

The approach of Gregoire and Schomer [3] is iterative aswell and it is mentioned that several algorithms are used forthe best solution in position and energy. If a certain thresh-old is reached, the simulation finishes with this solution. Acomparison to reality is made with one use case, a brakehose. No accuracy numbers are given, only a visual impres-sion and the description ”‘good fitting”’. The determinationof the material properties is done empirically. Gregoire andSchomer mention the problem of predeformation and sup-plementary torsion and relegate to construction tolerances.

Linn et al. [7] present an approach based on Kirchhoff’sgeometrical exact theory which is approximately solved byenergy minimization using a nonlinear conjugate gradientsmethod. It is mentioned that the algorithm is integrated in asoftware package and the real-time capability is comparableto [3]. The accuracy is neither mentioned nor proved.

3. Idea of Comparison

In this paper, the published approach of a high accuratereal time simulation of cables (fleXengine) [8, 5, 2] is com-pared with real cable setups. Since dynamic effects are dif-ficult to measure, the first step ist to take the quasistatic be-havior of our simulation tool into account. This is equiva-lent to the constraint that the system is at equilibrium aftereach time step and the energy is minimized.

In the virtual scene, the accuracy of a cable simulationsystem can only be judged by the users’ sense of plausi-bility and experience. For many applications, this solutionwith real-time interaction is already feasible and leads to areduction of prototypes. For further credibility, the accuracyof the simulation system has to be proved. In most cases, thecomparison was made with numerical solutions [18] or sim-ple curvature in reality (unpublished internal work). Com-plex curvature, gravity and torsional effects are not yet takeninto account.

Therefore we composed and constructed a testing setupfor complex cable and hose shapes (see fig. 2). Variablesetups and torsions can be applied to the specimen up to alength of 1000 mm. The position and orientation of bothsides is freely adjustable and on one side torsional forcescan be applied and measured. The cable specimen arebraided cables from 1mm2 to 10mm2, the bundle specimenhomogeneous and composite bundles with different tapinglike spiral taped, half taped, conduit and full taped. A handheld laser scanner digitizes the shape without influencingit at very high precision (0.1 mm resolution). The digitalshape is then imported in the simulation system and the cen-terline is extracted. The simulation system is embedded intothe Visual Decision Platform developed by IC:IDO GmbH1.The software is able to replace rigid geometry which repre-sents a cable or hose with a flexible object. The algorithmis able to reconstruct the round shape of the specimen veryprecisely, even if the laser scan is highly fragmented. Thusthe scanned object is replaced by a simulated object with ahigh number of supporting points (handles) to fit the origi-nal shape with very high precision. The internal procedureof this algorithm is confidential, the accuracy of this methodis seen in figure 5.

The next step is the determination of material propertiesof the real cable. Therefore the specimen is measured forstiffness (Young’s Modulus) and density. The stiffness isdetermined by measuring the force that is needed to benda defined length of the specimen for a given displacement.The stiffness measured by tensile test differs much from thebending test, which is due to the complex inner structure: abraided wire behaves in bending far softer than in the tensiletest, where it behaves almost like a rod. Thus the tensilestiffness is up to 100 times higher. Since in most cases the

1www.icido.com

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simulation is build to simulate bending behavior and shapes,the bending stiffness is the parameter of choice. Density ismeasured with a pycnometer. Additionally, inner and outerdiameter and the length of the specimen are measured.

With the scanned shape of the specimen and the materialproperties, the simulation can be performed. The end po-sitions of the scanned shape are taken and the tangents aremeasured. The end positions, the tangents, the diameters,the length, the material properties and the knowledge aboutthe direction of gravity are used as input parameters of thesimulation. Since the positions are taken out of the laserscan, the optimal case would now be a high match of thescanned and the simulated shape. The centerlines are ex-tracted and compared with NX2 to measure the deviationswith µm precision. The maximum deviation between thecenterlines in relation with the length of the specimen is de-fined as the error of the analysis.

Figure 2. The test setup for any possible com-bination of bending and torsion (a) with mea-surement system and scale for torque (b).The applied or risen torsion force can be readon the newton meter and related to the de-gree of rotation (c).

4. Precomputation of Material Properties

To increase the expressiveness of the comparison, thereal material properties have to be measured. This leads

2http://www.ugsplm.de/produkte/nx/

to several difficulties in the daily use of the tool.

1. Knowledge of components In early constructionstages, a rough routing of the cable is needed. Theknowledge of the final components is still dependentof the later usage of the bundle. Without the final con-struction idea, the bundle cannot be build and mea-sured.

2. Variants In the same product, the components of a ca-ble bundle differ depending of the fitting. The bundlecan contain e.g. three aerial cables or six or additionalsensor cables for e.g. inflation status. Thus, plenty ofvariants have to be measured.

3. Need of accuracy Not every part of a flexible com-ponent is critical regarding the shape. Motion, criticalobstacles and construction issues have to be handledwith high accuracy. If the environment is uncritical, a90%-solution might be adequate.

4. Cost-Benefit The measurements are costly and theconstruction of the cable takes critical time.

These considerations lead to the question if there is a fasterand cheaper way for some use cases, where e.g. a 90%-solution is adequate. Therefore investigations have beenmade to compute the material parameters of a bundle withonly the knowledge of the components. The material prop-erties of the components have to be measured as well, buttheir number is more limited than the combinations. For thesingle strands and their isolations the properties are knownand filed in a material database.

Known from beam theory [6], the bending stiffness (B)is given by the Young’s Module (E) and the moment of in-ertia of area (I) by

B = EI.

I is defined asI =

14π(R4

1 −R42),

whereR1 is the outer diameter andR2 the inner one regard-ing a hose. Since R2 is zero for cables, the equation for thebending stiffness reduces to

B =14πER4.

This equation is valid for homogeneous and circularcross sections. For composite cross sections, the center ofbending is no longer the center of the bundle since it is de-pendent of the material distribution. Thus two main axesof inertia are defined, which represent the minimum andthe maximum bending stress (see fig. 3, (a)). The crosssections are composed in an editor, where each color repre-sents a defined stiffness (see fig. 3, (b)). These directions

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are obtained via the computation of the eigenvalues of thebending stiffness matrix

Bij =(Bxx BxyByx Byy

), (1)

which is computed by

Bij =∫E(~s)(s− s0)i(s− s0)jdf. (2)

s and s0 are given in cross section coordinates. The mo-ments of inertia are related to the center of bending whichdiffers from the center of the bundle s0 and leads to the off-set s− s0.

For coaxial cross sections eqn 2 can be transfered to po-lar coordinates and can be rewritten as

B =∫πrErr

2dr. (3)

This leads to the known and proved analytical solutionfor coaxial cross sections (see fig. 3, (c))

B =∑i

14Eiπ(R4

1i −R42i). (4)

The torsional stiffness is described with the shear modu-lus µ. For composite cross sections, µ is no longer constantbut dependent on the position µ(x, y). Thus µ is partiallydifferenced and with the auxiliary function χ (see [6]), andthe substitutions ω(x, y) = µχ and α(x, y) = 1

µ we have

∂

∂x

(α∂ω

∂x

)+

∂

∂y

(α∂ω

∂y

)= −1. (5)

By taking into account the boundary conditions for the edgehandling (ω = 0), the torsional stiffnessC can be computedvia the elastic torsional energy. For round inhomogeneouscross section we obtain

C = 4∫ (

α(x, y) · (∇ω(x, y))2)df. (6)

For coaxial cables, the substitution and auxiliary func-tion from eqn 5 depend only on the radius r of the crosssection. Thus the equation is reformulated with the Laplaceoperator for twodimensional polar coordinates as

d

dr

(αdω

dr

)+

1r

(αdω

dr

)= −1. (7)

With the substitution λ = α(dω/dr) the equation becomesan ordinary differential equation of first order and can bewritten as

λ′ +1rλ = −1. (8)

Figure 3. A typical cross section of a bundle(a) and the eigenvalues with maximum andminimum bending stiffness visualized by thelength of the vector (b). Typical coaxial ca-bles (c).

The solution λ = − 12r is inserted in eqn 6 and leads to

the equation for torsional stiffness of coaxial cables

C =12π∑i

µi(R4i1 −R4

i2). (9)

5. Results

The cable and bundle specimen are chosen from fre-quently exerted use cases. This increases the expressivenessof the tests since the results can be compared to the realprototype. 72 measurements are performed with the setupsshown in figure 5 with and without torsion. The setup isbuild with least predeformation possible, though preformedbehavior is common in manual bundle fabrication and trans-portation. The steps of the comparison are demonstrated infigure 5. The error is defined as the maximum deviation inrelation to the length of the specimen (100%). The resultsare shown in figure 5.

The measurements of the stiffness are done with severalbending tests of the same sort of cable or bundle. They aremonitored and a graph is drawn with the force in relationto the bending distance. This graph is a hysteresis and nostraight line. This means that even for small deflections, thebehavior of the specimen is not purely elastic. Additionally,the bundles showed the behavior of inner displacements andfriction.

The computation of the material properties shown in sec-tion 4 is mathematically correct. Deviances occur due to theplastic behavior of the real specimen, whereas in the equa-tions plasticity is neglected. Furthermore, effects like thetaping and friction are not taken into account which leadsto deviances as well. This explains the discrepancies of themeasurement of a specimen and the computed value, whichdiffer about the factor of two. The computation and mea-surements of two bending stiffnesses showed small vari-ances depending of the bending direction. E.g. for the

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Figure 4. Three setups used for the valida-tion. The test were performed with no torsionand 45◦ or 90◦ torsion.

bundle shown in figure 3, the maximum measured bend-ing stiffness is 0,034158 Nm2 and the minimum 0,028351Nm2.

The measured shape of the composite bundles lead to theassumption that the effects of different bending stiffnessesdepending on the bending direction are negligible. Due tointernal twist and the small differences in the values, theshape behaves like a homogeneous or coaxial specimen.

6. Conclusion

The results show the very high accuracy of the simula-tion in VR with an average precision higher than 99% forthe 72 tests. The bending and torsion of cables and bundlesis simulated physically correct. The main source of errorare material variances due to manual fabrication which leadto preformed object behavior. The accuracy is much higherthan the constraints usually given by the client, so the simu-lation is able to replace physical mock-ups. During the tests,much knowledge about measurement and material proper-ties was gathered and with these experiences, further testscan be performed very efficiently.

In this paper it is shown that the material distribution

Figure 5. The laser scan of the specimen (a),the reconstruction of the centerline of thespecimen (b, c), the simulation result (d) andthe comparison with NX.

Figure 6. From 72 specimen, the error of 50test is below 1%. The main source of er-ror (3% and 4%) is related to predeformation.This proves the high accuracy of the simula-tion and the correctness of the measurementof the material properties.

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leads to differing bending stiffnesses and that the center ofgravity changes in the cross section. Concerning the simu-lation, we proved that this effect nullifies in reality, becausethe distribution changes over the length of the specimen dueto internal twist. This behavior is another demonstration ofthe validity of the approach: one bundle, one material.

In the approach of computing the bending and torsionalstiffness with the knowledge of the cross section and thecomponent parts, further physical issues have to be takeninto account. Elastic-plastic effects, the inner friction andthe way of taping have high influence on the combinedproperties of the specimen. This sector will be of high in-terest for further research projects.

7. Acknowledgments

Thanks to Delphi Packard for specimen and funding.

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