Here we derive analytic expressions for the scalar parameters which appear in the generalized Euler decomposition of the rotational matrices in R 3. The axes of rotations in the decomposition are almost arbitrary and they need only to obey a simple condition to guarantee that the problem is well posed. A special attention is given to the case when the rotation is decomposable using only two rotations and for this case quite elegant expressions for the parameters were derived. In certain cases one encounters infinite parameters due to the rotations by an angle π (the so called half turns). We utilize both geometric and algebraic methods to obtain those conditions that can be used to predict and deal with various configurations of that kind and then, applying l'Hôpital's rule, we easily obtain the solutions in terms of linear fractional functions. The results are summarized in two Tables and a flowchart presenting in full details the procedure. Contents 1 Introduction 60 2 The Generic Case 64 3 The Symmetric Case 69 4 Decomposition Into Two Rotations 71

The paper discusses manipulator modelling and control through a nonstandard parametrization of rotation motions. The advantage of the method is the computational facilities arising at the kinematical level, not from efficient presentation, but from fundamental topological considerations of the configurational manifold provided with a Lie group structure.

No abstract

Abstract. In this paper, a diversity of problems arising in manipulator modelling and control are treated on the group configurational manifold Qct in place of the standard configurational space Q. This possibility is a result of the vector-parametrization of the SO(3) Lie group. The simple composition law of vector parameters as well others of their nice properties reduce the computational burden (in comparison with the methods used up to now) in solving direct and inverse kinematic problems, both in dynamic modelling and full simulation of the motion of a manipulator system. This fact, proved in relation with the standard configurational space approach, becomes stronger over the group manifold where all kinematical equations are pure algebraic and the differential equations of motion 'feel' the group structure. The suggested approach gives the opportunities for the powerful methods of Lie group theory to be involved in controlability and observability of the treatment of manipulator systems. The introduction of a dual vector parameter provides an interesting interplay of special geometrical considerations and special algebraic structures in this important area of the application of linear algebra.

No abstract

This work is a review of our research activity during the last ten years concerning the problems of modeling and control of multi-body mechanical systems. Because the treatment of the above topics is quite sensitive with respect to the different parameterizations of the rotation group in three dimensional space SO(3) and because the properties of the parameterization more or less influence the efficiency of the dynamic model, here the so called vector-parameter is used for parallel considerations. The consideration of the mechanical system in the configurational space of pure vector-parameters with a group structure opens the possibilities for the Lie group theory to be applied in the problems of the dynamics and control. The sections in this paper present independent parts of an unified scientific approach.

We use a vector parameter description of the Lorentz groups in R 2,1 and R 3,1 to obtain an exact expression for the Thomas factor as a geometric phase. The effect of phase accumulation in Thomas-Wigner precession phenomena is seen as a manifestation of the hyperbolic solid angle theorem. On the infinitesimal level, our description involves affine connections on the noncompact Hopf fibrations U(1) → SU(1, 1) → Δ and SU(2) → PSL(2, C) → H 3 . The associated gauge field is a restriction of the familiar Yang-Mills anti-instanton. We also consider the dual compact case, and we discuss generalizations to arbitrary dimensions and applications in various branches of theoretical physics.

No abstract