Abstract: This paper presents a method for studying multibody systems subjected to non-linear non-holonomic constraints. The method employs a modified form of Kane's equations and a differentiated form of the constraint equations. Kane's equations are modified by the inclusion of generalized constraint forces, and the differentiated constraint equations become linear in higher derivatives of the generalized coordinates. It is observed that the generalized constraint force array is proportional to the transpose of the matrix of coefficients of the constraint equations. Using this relationship the governing equations are reduced to a consistent set of differential equations through a multiplication with an orthogonal complement of the constraint equation matrix. The method is illustrated with the classical Appell-Hamel problem.

Abstract: In this paper a unified numerical theory is presented for a lar e class of recent multibody dynamics formula! ti on^^.^. J0J3J4J6 . It is shown that, although the derivation and presentation of many of these methods are quite dissimilar, they may all be represented concisely using Maggi's dynamical equations and common projective methods from linear operator theory. These methods all eliminate constraint contributions from the governing equations. It is shown that this reduction process amounts to projecting the system of governing differential-algebraic equations to a "constraint-free" subspace. The process of equation reduction and constraint force recovery is achieved by generating an orthogonal basis for the nullspace of the transposed constraint matrix, and its orthogonal complement, the range of the transposed constraint matrix. By explicitly deriving these dual bases for several typical methods in the class, it is shown that the methods differ only in the particular basis selected for the nullspace and its orthogonal complement. This enables one to easily'compare the numerical efficiency of the various methods, as well as their stability properties. Furthermore, the unified theory enables one to derive concurrent multiprocessing approaches applicable to a wide combination of simulation methodshardware architectures. MAGGI'S EQUATIONS OF MOTION Maggi's equations were first presented in 1896, and again in 1901. Although the method is conspicuously absent from most texts in the English language, Neimark and ~ufaev" summarize the essentials of Maggi's approach. Until quite recently, the method has not been amenable to actual calculations and has been of (little) academic interest only. However, to obtain practical implementations of the method, it is necessary to extend the original, classical definition of the method only slightly using common projective techniques from linear algebra. As shall be shown in this section, the advent of digital computers now permits a wide variety of practical numerical implementations of the approach. An axiomatic development of Maggi's equations is beyond the scope of this paper, consequently the interested reader is referred to either ~eimark" or ~urdila*. As a practical matter, on can obtain Maggi's equations by starting with Lagrange's equations in variational form subject to the variational constraints This equation can be alternatively represented in terms of aDxNmatrixaas This equation can now be interpreted as requiring that the N-vector 6q lie in the nullspace of the time-varying, linear operator a, or that 6q be orthogonal to the range of aT. The range of aT is equal to the span of the columns of aT, and is consequently of dimension D if the constraints are independent. Deferring for the moment their calculation, let [Al, A2, ... AI] be any basis for the nullspace of the constraint matrix a. The vector 6q may lie anywhere within the span of [Al, AZ, ... A& In other words, it is true that

Abstract: A method for detecting and identifying errors in sensors for state variables which are linked to measurement variables, directly supplied by the sensors, by means of a measurement equation m = Hx + omega , where m is a vector of the measurement variables, x is a vector of the state variables and H is the measuring matrix and the order of m is greater than the order of x, contains the following method steps: a) Determination of validation vectors vi as column vectors of a matrix P = [v1v2...vn] b) determination of a parity vector v as element of a parity space which is the orthogonal complement of the signal space as linear combination of the validation vectors with the associated elements m of the measurement vector m: c) formation of a detection function DF = as a scalar product of the parity vector by itself and check whether this detection function is greater than or less than a predetermined limit value, the transgression of this limit value signalling the existence of an error, d) in the presence of an error, calculation of locating functions from the components of the parity vector and determination of the maximum locating function from which the faulty sensor can be derived e) reconfiguration of the sensor signals, taking into consideration the error status thus determined.

Abstract: A brief survey on different methods to examine the controllability of bilinear systems is given. Some results, obtained with the help of group theory, on the controllability of strictly bilinear systems and homogeneous-in-the-state bilinear systems are presented in more detail. Finally homogeneous-in-the-state bilinear systems in state spaces of dimensions two and three are discussed. The cases where the system matrices of the homogeneous-in-the-state bilinear system generate a Lie algebra which spans a space with a dimension equal to the dimension of the state space are considered, i.e. the corresponding strictly bilinear system is controllable. For the two-dimensional case it turns out that there are only two situations when the strictly bilinear system is controllable but the homogeneous-in-the-state bilinear system is not. By checking the vector fields on the boundaries of the reachable sets obtained in these two situations, one can easily determine whether the homogeneous-in-the-state bilinear system is controllable or not, provided the Lie algebra of the system matrices spans R2. This approach is extended to the three-dimensional case and thus, previously obtained results can be generalized.

Abstract: A method for determining the states of the hidden units of feedforward neural networks for an arbitrary output function is proposed. The method uses properties of a vector space spanned by state vectors. The state vector used represents a set of states (input states, inner states, or output states) of a unit for many sample data. The inner state vector is the linear combination of the input state vectors and is nonlinearly transformed into the output state vector. Internal representation can be expressed by the output state vectors for the hidden units, which are the input state vectors necessary for the output unit. The problem is how to determine the appropriate internal representation in order to produce an arbitrary output function. This method, called the orthogonal complement method (OCM), is based on the orthogonality between the subspace spanned by the input state vectors and its orthogonal complement. The nonlinearity of the vector transformation is treated as a constraint with respect to the inner state vector. Unknown components of the output state vectors for the hidden units can be determined from this orthogonality, and the number of necessary hidden units can be estimated from the dimension of the subspace spanned by the input state vectors. The OCM is very useful for binary output systems, since this calculation can be simplified by using a basis of the orthogonal complement. Using a basic procedure of the OCM, a minimum network can be designed for about 70% of all four-variable Boolean functions

Abstract: In this text we present unpublished results by Eugeniusz Kusak and Wojciech Leonczuk. They contain an axiomatic description of the class of all spaces 〈V ; ⊥ξ〉, where V is a vector space over a field F, ξ : V × V → F is a bilinear symmetric form i.e. ξ(x, y) = ξ(y, x) and x ⊥ξ y iff ξ(x, y) = 0 for x, y ∈ V . They also contain an effective construction of bilinear symmetric form ξ for given orthogonal space 〈V ; ⊥〉 such that ⊥=⊥ξ. The basic tool used in this method is the notion of orthogonal projection J(a, b, x) for a, b, x ∈ V . We should stress the fact that axioms of orthogonal and symplectic spaces differ only by one axiom, namely: x ⊥ y+ez&y ⊥ z+ex ⇒ z ⊥ x+ey. For e = −1 we get the axiom on three perpendiculars characterizing orthogonal geometry. For e = +1 we get the axiom characterizing symplectic geometry see [1].

Abstract: The key questions in the scattering theory for time-periodic potentials are as follows: 1) the uniform boundedness of the global energy; 2) the local energy decay.