Interactive simulation of one-dimensional flexible parts

Abstract

In this paper, we present a system for simulating one dimensional flexible parts such as cables or hose. The modelling of bending and torsion follows the Cosserat model. For this purpose we use a generalized spring–mass system and describe its configuration by a carefully chosen set of coordinates. Gravity and contact forces as well as the forces responsible for length conservation are expressed in Cartesian coordinates. But bending and torsion effects can be dealt with more effectively by using quaternions to represent the orientation of the segments joining two neighbouring mass points. This augmented system allows an easy formulation of all interactions with the best appropriate coordinate type and yields a strongly banded Hessian matrix. An energy minimizing process accounts for a solution exempt from the oscillations that are typical of spring–mass systems. The whole system is numerically stable and can be solved at interactive frame rates. It is integrated in a virtual reality software for use in applications such as cable routing and assembly simulation.

Introduction

The automotive industry aims to reduce development costs and time while meeting the demand for quality and for an increasing model range. In order to meet this challenge, more and more work is done digitally. Styling reviews, digital mock-ups and assembly simulations are used at an early stage in the development process. Thus, potential problems can be detected and solved much earlier, long before the first physical prototype is built. Assembly simulation is one of the applications used in the construction design process: the virtual prototype is tested and optimized for feasibility and ease of assembly. Like in other virtual reality applications, the physical behaviour of the components needs to be simulated: collisions must be detected and contact forces must be calculated to impede the interpenetration of objects and to make them slide on each other when collisions occur. Our work is based on the virtual reality software veo, developed by DaimlerChrysler Research and Technologies, which already has a real-time collision detection and interactive contact simulation [1] as well as a real-time multibody dynamics [2]. This software had to treat flexible parts as rigid bodies, however, because it did not have a realistic way of dealing with flexible bodies. This implies that deformations that can occur in the physical world cannot be simulated, which limits the possibilities of the tool. A typical example of this would be an assembly simulation in which a cable should be pushed slightly aside to permit the mounting of another part. Studies from the business units show that most of the problems concerning flexible parts are encountered with cables, wire harnesses and hoses. These parts are also involved in another particular application: the routing. On the one hand, wire harnesses are becoming more and more complex as the use of electrical and electronic components in vehicles grows; on the other hand, they often need to be modified to accommodate for changes in surrounding parts or for optimization. To meet the interests of the business units, we have chosen to first concentrate on simulating these parts since they cover the most urgent need. For these bodies, one of their dimension (the length) is much bigger than the other two and their centreline contains most of the information needed to represent them. This one-dimensional nature leads to simplifications in the simulation compared to other objects such as flexible surfaces or bodies.

Several approaches have been used for simulating various types of flexible bodies. We focus in particular on one-dimensional bodies, where the spectrum of solutions ranges from purely graphical representation of oscillations without any physics and with very low computational requirements [3] to a complete finite element simulation, such as has been used to predict the mechanical properties of cloth [4]. The domains which have contributed to the simulation of flexible parts are as diverse as the flexible parts themselves: hair and cloth simulation (see [5], [6] for a survey on hair simulation and [7] for a survey on cloth simulation) for use in computer graphics, thread simulation for surgical training or laparoscopy, assembly simulation tools... Among the most popular computational models are differential equations [8], chains of rigid bodies [9] or spring–mass systems [10]. Mass-spring systems are a versatile way of simulating deformations and have been used for simulating a variety of flexible parts. [11] use a triangular mesh for modelling cloth. [10] implements a cable as a spring–mass model with stiff linear springs for the length conservation and torsion springs (the energy is proportional to the angle between two segments) for the bending. Hergenröther [9] models a cable as a chain of cylinder segments connected by ball joints. The chain at first has two segments and is iteratively refined, doubling the number of segments at each iteration, thus giving an inexact but fast dynamic simulation and a more exact, slower static one. [12] implements an impulse-based simulation for cables: impulses are applied at the beginning of each time-step for ensuring that the constraints are enforced in the rigid-body chain. [13] uses geometrically exact dynamic splines for modelling cables. Suturing ropes are simulated in [14] with a Follow the Leader algorithm: after a node has been moved to its new position, its neighbours are then moved along the line joining their old position to the new position of the moved node so that they are at the correct distance from the grasped node. The other nodes are then moved iteratively. In the case of contradictory moving directions (there are two grasped nodes, or collisions...), the resulting displacement is an average of the displacement induced by each of the constraints, which do not necessarily preserve the constraints, but the inaccuracies are not visually noticeable and are averaged out after a few iterations.

Few approaches deal with torsion. This is unfortunate since, as shown for example in [15], torsion plays a crucial part in the deformation of a cable. One of the most interesting approaches taking the torsion into account is the one from Pai [8]. He uses a Cosserat model as a base for the simulation of suture strands during laparoscopic surgery. These strands are objects visually well approximated by a curve but nevertheless present three-dimensional body properties like twisting. In the typical use case in surgery simulation, the position and direction of the strand are defined at one end (corresponding to the end fixed in human tissues) and the forces and moments are defined at the other end, corresponding to the needle haptic device. The ordinary differential equation resulting from the Cosserat model is integrated in two passes to calculate all the variables. Another implementation of this model can be found in [16], [17] for modelling hair. Each master strand is discretized as a smooth sequence of helical segments. The variables are then the constant curvatures and torsions of those segments. The shape of the strand can be calculated by integration. The system can be solved either by energy-minimization [16] or dynamically with a Lagrangian formulation [17]. A similar formulation, using also the curvatures and torsion as variables can be found in [18] for path-planning techniques.

Being aware of the importance of torsion for the deformations of a cable, it became one of our main interests. After several attempts, we have finally chosen a generalized spring–mass system for modelling the cables. This system uses a mixed coordinate system that contains at the same time ordinary position coordinates and quaternions representing the orientation of segments on the basis of which the torsion is easy to calculate. For enforcing constraints like the conservation of length, we introduce an integral force analogous to the action of a proportional–integral controller.