We present a new proof and extension of the classical Sylvester Inertia Theorem to a pair of non-Hermitian matrices which satisfies the property that any real linear combination of the pair has only real eigenvalues. In the proof, we embed the given problem in a one-parameter family of related problems and examine the eigencurves of the family. The proof requires only elementary matrix theory a...

In recent years there has been interest in a theorem on positive definite matrices known as Lyapunov's theorem. Several authors have proved generalizations of this theorem, (WIELANDT [29J, TAUSSKY [24J, [25J, [26J , OSTROWSKISCHNEIDER [20J, GIVENS [10J, CARLSON-SCHNEIDER [3J, CARLSON [4J) . Lyapunov's theorem and its generalizations have become known as inertia theorems. In this note we shall u...

We propose a novel unifying scheme for parallel implementation of articulated robot dynamics algorithms. It is based on a unified Lie group notation for deriving the equations of motion of articulated robots, where various wellknown forward algorithms differ only by their joint inertia matrix inversion strategies. This new scheme leads to a unified abstraction of state-of-the-art forward dynami...

Sign patterns consisting of some positive and some negative columns, with at least one of each kind, are shown to allow any self-conjugate spectrum, and thus to allow any inertia. In the case of the n× n sign pattern with all columns positive, given any self-conjugate multiset consisting of n−1 complex numbers supplemented by a sufficiently large positive number, it is shown how to construct a ...

For a real matrix A whose spectrum avoids the imaginary axis, it is shown that the three following, seemingly independent problems, are in fact equivalent. • Characterizing the set of real symmetric solutions of the algebraic Lyapunov inclusion associated with A. • The image of all Nevalinna-Pick interpolations associated with the spectrum of A. • The structure of the Convex Invertible Cone gen...

A sign pattern is a matrix with entries in {+,−, 0}. A full sign pattern has no zero entries. The refined inertia of a matrix pattern is defined and techniques are developed for constructing potentially nilpotent full sign patterns. Such patterns are spectrally arbitrary. These techniques can also be used to construct potentially nilpotent sign patterns that are not full, as well as potentially...

A correction to the previously published article “New Form of Kane’s Equations of Motion for Constrained Systems”1 is presented. Misuse of the transformation matrix between time rates of change of the generalized coordinates and generalized speeds (sometimes called motion variables) resulted in a false conclusion concerning the symmetry of the generalized inertia matrix. The generalized inertia...

We develop a method to simplify the partial differential equations (PDEs) associated to the potential energy for interconnection and damping assignment passivity based control (IDA-PBC) of a class of underactuated mechanical systems (UMSs). Solving the PDEs, also called the matching equations, is the main difficulty in the construction and application of the IDA-PBC. We propose a simplification...

Mark W. Spong textbook [13] for background of the robotics application of this technique. The novelty of our result in the present paper lies in the fact that the uncertainty bounds needed to derive the control law and to prove uniform ultimate boundedness of the tracking error depend only on the inertia parameters of the robot. Previous applications of the Leitmann approach to robot manipulato...