Interval Algorithms in Modeling of Multibody Systems

Interval Algorithms in Modeling of Multibody Systems Ekaterina Auer, Andr´es Kecskem´ethy, Martin T¨ andl, and Holger Traczinski University of Duisburg-Essen, Germany, {auer, traczinski}@informatik.uni-duisburg.de {a.kecskemethy, m.taendl}@uni-duisburg.de

Abstract. We will show how a variety of interval algorithms have found their use in the multibody modeling program MOBILE. This paper acquaints the reader with the key features of this open source software, describes how interval arithmetic help to implement new transmission elements, and reports on interval modeling of dynamics, which is an inherent part of multibody simulations. In the latter case, the interval extension of MOBILE enhanced with an interval initial value problem solver (based on VNODE) is presented. The functionality of this application is shown with some examples. We provide insights into techniques used to enhance already existing modeling software with interval arithmetic concepts.

1

Interval Arithmetic in MOBILE: Areas of Application and Integration Strategies

Interval arithmetic is often criticized for its inapplicability to real life problems. This work claims the contrary by showing how it can be employed in multibody systems’ modeling, an important area of applied physics, and in particular, in the program MOBILE. Interval arithmetic is used here to not only ensure the validity of the obtained results, but to also provide new modeling opportunities. Mechanical interactions are usually modeled with the help of differential equations. It would be very time consuming to manually make up these equations each time. For that reason various types of modeling software have found a market in industry. Usually, such software produces the respective system of differential equations from the (formalized) description of an arbitrary mechanical system and is also capable of solving it thus characterizing the necessary system’s properties. In the present context, we employ the multibody library MOBILE described in [1, 2]. It is able to model arbitrary mechanical systems and is characterized by its high computational speed (section 2 of this paper describes this program in more detail). In the process of solving different problems with MOBILE, new tasks presented themselves, some of which proved to be most effectively dealt with by applying interval techniques. As a simple example of such a task, the incorporation

of some external measurements as parameters into a model can be considered. Measurements are usually performed with a (small) error, the influence of which on the system’s behavior is sometimes of interest. Moreover, the models always differ, if only slightly, from real life systems. Hence, it is useful to allow some uncertainty in the model and see how it affects the results. Intervals offer an elegant way for solving the above tasks. Thus, it is appropriate to combine interval principles with modeling algorithms, which presupposes integration of interval methods into the already existing program MOBILE. This integration is performed in three layers. The basic layer is the interfacing between MOBILE and interval arithmetic; the package Profil/Bias [3] was chosen to provide the appropriate data types and methods. Based on this interface, additional structures need to be defined. For example, a simple replacement of floating-point arithmetic with interval arithmetic is insufficient because of its undesirable by-products such as the wrapping effect. Therefore, we have to improve the “naive” interval extension by exploiting knowledge about underlying MOBILE structures. Once this is done, the interval extension can be enhanced with more complicated algorithms, for example, for solving ordinary differential equations (ODEs) or computing validated distances, as well as design new MOBILE components, which allow, for example, uncertainty in measurements. All that constitutes the middle layer of integration, on top of which a connection to the outside world can be considered. Thus, the third and last integration layer require building interfaces to industrial modeling software [4]. Our goal is to implement an extension of MOBILE capable of interval calculus. To achieve this, we will proceed on two levels: implementation of interval kinematics and interval dynamics of mechanical systems. To develop the former, basic interval algorithms, such as addition, subtraction, etc. are required, as well as more complicated ones, such as solution of interval constraint equations, etc. The present state of this side of implementation is reflected in section 3 of this paper. To implement interval dynamics, one has to find an interface between interval initial value problem solvers (IIVPS) and MOBILE. The task of integrating an IIVPS into certain types of modeling software is not completely free from difficulties. One of the major problems is obtaining derivatives. On the one hand, there are several interval algorithms to solve initial value problems. Their common feature is the presence of several system function’s derivatives. As a rule, the higher their order, the tighter the enclosure obtained. The well-known derivative free methods from numerics (Adams, Runge-Kutta, etc.) proved themselves hard to adapt to intervals. On the other hand, most of the modeling software has no facilities to produce derivatives of arbitrary order. The usual methods of automatic differentiation, employed in many IIVPS, are impossible to make use of, because they require the right hand side of a given problem to be symbolically expressed, while in most cases this expression remains unknown. All the information given about the system function is its “numerical” values at some arbitrary points and its

algorithmic representation in a certain programming language. Therefore, the additional task to be solved on this implementation level is obtaining derivatives, which comply with the demands of validated algorithms, using only the above information. Possible ways of dealing with this problem as well as achievements towards modeling of systems’ dynamics are described in section 4 of this paper. A short summary of the most important results and a prospect on further work can be found at the conclusion of this paper.

2

The Multibody Modeling Library MOBILE

The modeling and simulation of the dynamical behavior of mechanical systems is a well-studied field in mechatronics. During the last 25 years, a large number of researchers have developed several formalisms for the automatic generation and resolution of the dynamical equations of multibody systems [5]. Some of these methods are still used today as universal engines for mechanics-based calculations in modern CAD systems, including for example ADAMS [6] in IDEAS and SD-FAST [7] in Pro/ENGINEER. Other formalisms have concentrated on specific areas of engineering, including for example recursive methods [8, 9] for robotics, or symbolic computation methods for real time applications [10, 11]. These approaches have the advantage of being comprehensive and provide comfortable user interfaces. However, due to their monolithic structure they lack the efficiency and capability of interaction with other simulation packages. The present approach uses object-oriented programming for defining an openarchitecture multibody library. The mechanical components are modeled as abstract mappings, termed kinetostatic transmission elements, which transmit motion and loads between sets of input and output variables called state objects. This results in an intuitive formulation, which allows the designer to put together the models of the parts of a mechatronic system in virtually the same way as they would actually be assembled in the real world. Moreover, by substituting other mathematical objects for real numbers, generic multibody formulations can be obtained which can be used for example for interval, stochastic, and fuzzy analysis. The multibody library MOBILE was implemented using the object-oriented programming language C++. Currently, only rigid bodies are modeled, but extensions to problems of structural mechanics, hydraulics and control theory can be incorporated into the general procedure. Mathematically, the operations relating to the kinetostatic transmission elements correspond to well-known mappings of differential geometry: the transmission of position and velocity correspond to a nonlinear mapping between two smooth manifolds and the corresponding push-forward function for tangent vectors, while force mapping corresponds to the pull-back function being applied to cotangent vectors. From a computational point of view, the applied method renders a responsibility driven client/server model [12] in which multibody operations are defined as “services” provided by an object at any time during program execution independently of its internal implementation according to a specific “contract”.

In MOBILE, the basic “contract” of kinetostatic transmission elements consists of two main services: one for transmission of motion (“doMotion”) and one for transmission of forces (“doForce”). More elaborated objects are defined at the following three levels of modeling complexity: (1) basic modeling, which involves only pure kinetostatic transmission elements, (2) sparse-Jacobian modeling, in which the interconnection structure and efficient methods for obtaining velocity transformations are considered, and (3) inertia-transmission modeling, in which the individual components are regarded as Riemannian manifolds able to generate and transmit mass properties. A description of the latter two levels of modeling complexity can be found under [1] and [13]. 2.1

The Concept of Kinetostatic Transmission Elements

The central modeling element for mechanical systems is the kinetostatic transmission element (Fig. 1), which regards a mechanical component as an element MoMap that maps a set of n scalar variables collected in the input vector q to a set of m scalar variables collected in the output vector q 0 .

q

-

q˙

-

q¨ Q

- q0 - q˙ 0

“map”

- ¨q 0



Q0

Fig. 1. Simple model of a kinetostatic transmission element

Associated with this mapping, there exist three kinematic functions and a force-associated function. The kinematic functions are the mapping itself and its first and second derivatives. These are collected in the motion transmission functions    position: q0 = ϕ ( q )   0 . (1) velocity: q˙ = Jφ q˙    0  q + J˙φ q˙ acceleration: ¨q = Jφ ¨

Here, Jφ = ∂ϕ/∂q is the m × n Jacobian of the transmission element, which is not required explicitly by the clients of the MoMap element. For the force transmission function, one assumes that the transmission element is ideal , i. e. that it neither consumes nor produces power. Then, virtual work at the input and output are equal: T δq T Q = δq 0 Q0 . (2) After substituting δq 0 = Jφ δq and noting that this condition must hold for all virtual displacements δq ∈ Rn , one obtains force:

Q = JφT Q0 ,

(3)

where JφT is the transpose of the Jacobian Jφ . This transformation is directed from the (kinematical) output of the transmission element to its (kinematical) input. Note also that, in general, Jφ need not be regular, in fact, not even square, so one cannot assume that (3) can be inverted. Thus force transmission is in general directed in the opposite direction to motion transmission. In MOBILE, each transmission element “remembers” its once defined inputs and outputs for its lifetime. Hence, execution of the “doMotion” and “doForce” is possible by linking dynamically and without any arguments. Moreover, kinetostatic transmission elements can be concatenated by connecting the outputs of one element to the inputs of the other. The transmission functions of such a composite transmission element (termed MoMapChain in MOBILE) can be realized by concatenation of motion transmission in the order of the mechanical chain starting at the inertial system, and in reverse order for force transmission. In MOBILE, MoMapChain objects are simply ordered lists supporting the “