32,485 research outputs found
Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters
The paper develops Newton's method of finding multiple eigenvalues with one
Jordan block and corresponding generalized eigenvectors for matrices dependent
on parameters. It computes the nearest value of a parameter vector with a
matrix having a multiple eigenvalue of given multiplicity. The method also
works in the whole matrix space (in the absence of parameters). The approach is
based on the versal deformation theory for matrices. Numerical examples are
given. The implementation of the method in MATLAB code is available.Comment: 19 pages, 3 figure
A spatial impedance controller for robotic manipulation
Mechanical impedance is the dynamic generalization of stiffness, and determines interactive behavior by definition. Although the argument for explicitly controlling impedance is strong, impedance control has had only a modest impact on robotic manipulator control practice. This is due in part to the fact that it is difficult to select suitable impedances given tasks. A spatial impedance controller is presented that simplifies impedance selection. Impedance is characterized using ¿spatially affine¿ families of compliance and damping, which are characterized by nonspatial and spatial parameters. Nonspatial parameters are selected independently of configuration of the object with which the robot must interact. Spatial parameters depend on object configurations, but transform in an intuitive, well-defined way. Control laws corresponding to these compliance and damping families are derived assuming a commonly used robot model. While the compliance control law was implemented in simulation and on a real robot, this paper emphasizes the underlying theor
Quasi-periodic stability of normally resonant tori
We study quasi-periodic tori under a normal-internal resonance, possibly with
multiple eigenvalues. Two non-degeneracy conditions play a role. The first of
these generalizes invertibility of the Floquet matrix and prevents drift of the
lower dimensional torus. The second condition involves a Kolmogorov-like
variation of the internal frequencies and simultaneously versality of the
Floquet matrix unfolding. We focus on the reversible setting, but our results
carry over to the Hamiltonian and dissipative contexts
Miniversal deformations of matrices of bilinear forms
V.I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a
miniversal deformation of matrices under similarity; that is, a simple normal
form to which not only a given square matrix A but all matrices B close to it
can be reduced by similarity transformations that smoothly depend on the
entries of B. We construct a miniversal deformation of matrices under
congruence.Comment: 39 pages. The first version of this paper was published as Preprint
RT-MAT 2007-04, Universidade de Sao Paulo, 2007, 34 p. The work was done
while the second author was visiting the University of Sao Paulo supported by
the Fapesp grants (05/59407-6 and 2010/07278-6). arXiv admin note:
substantial text overlap with arXiv:1105.216
A general approach to transforming finite elements
The use of a reference element on which a finite element basis is constructed
once and mapped to each cell in a mesh greatly expedites the structure and
efficiency of finite element codes. However, many famous finite elements such
as Hermite, Morley, Argyris, and Bell, do not possess the kind of equivalence
needed to work with a reference element in the standard way. This paper gives a
generalizated approach to mapping bases for such finite elements by means of
studying relationships between the finite element nodes under push-forward.Comment: 28 page
Transformation to versal deformations of matrices
AbstractIn the paper versal deformations of matrices are considered. The versal deformation is a matrix family inducing an arbitrary multi-parameter deformation of a given matrix by an appropriate smooth change of parameters and basis. Given a deformation of a matrix, it is suggested to find transformation functions (the change of parameters and the change of basis dependent on parameters) in the form of Taylor series. The general method of construction of recurrent procedures for calculation of coefficients in the Taylor expansions is developed and used for the cases of real and complex matrices, elements of classical Lie and Jordan algebras, and infinitesimally reversible matrices. Several examples are given and studied in detail. Applications of the suggested approach to problems of stability, singularity, and perturbation theories are discussed
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