Petr Vašík

Notes on Differential Kinematics in Conformal Geometric Algebra Approach

We consider different elements of a 5D conformal geometric algebra (CGA) as moving geometric objects whose final position is given by a specific kinematic chain. We show the form of the differential kinematics equations for different CGA elements, in particular point pairs, spheres and their centres.

Tangent structures: sector-forms, jets and connections

To define a higher order connection on a fibered manifold one can use the sections of nonholonomic jet prolongations. However, a more natural approach seems to be the one assuming the structure of a higher-order tangent bundle and using Whites sector-forms on these bundles.

Dual numbers approach in multiaxis machines error modeling

Multiaxis machines error modeling is set in the context of modern differential geometry and linear algebra. We apply special classes of matrices over dual numbers and propose a generalization of such concept by means of general Weil algebras. We show that the classification of the geometric errors follows directly from the algebraic properties of the matrices over dual numbers and thus the calculus over the dual numbers is the proper tool for the methodology of multiaxis machines error modeling.

Nilpotent approximation of a trident snake robot controlling distribution

We construct a privileged system of coordinates with respect to the controlling distribution of a trident snake robot and, furthermore, we construct a nilpotent approximation with respect to the given filtration. Note that all constructions are local in the neighbourhood of a particular point. We compare the motions corresponding to the Lie bracket of the original controlling vector fields and their nilpotent approximation.

DUAL NUMBERS ARITHMETIC IN MULTI AXIS MACHINE ERROR MODELING

and thus more convenient setting can be used. As algebraically the properties of , , xx yx zx ň ň ň coincide we can consider these as the coefficient of one element only and thus we can work in the dual numbers algebra, see [Hrdina 2014]. Note that this simplification brings minor complications, particularly the number of remaining terms increases, but, on the other hand, the question of the geometric interpretation of these higher order error terms is also interesting. For instance, one can distinguish the rotation error w.r.t. the x and y axis while translating along the axis x . Let us recall the definition of the dual numbers. It is a set { | , } a b a b     D (3) endowed with the operations summation and multiplication

A Note on Jet and Geometric Approach to Higher Order Connections

We compare two ways of interpreting higher order connections. The geometric approach lies in the decomposition of higher order tangent space into the horizontal and vertical structures while the jet-like approach considers a higher order connection as the section of a jet prolongation of a fibered manifold. Particularly, we use the Ehresmann prolongation of a general connection and study the result from the point of view of geometric theory. We pay attention to linear connections, too.

Semiholonomic Second Order Connections Associated with Material Bodies

The thermomechanical behavior of a material is expressed mathematically by means of one or more constitutive equations representing the response of the body to the history of its deformation and temperature. These settings induce a set of connections which can express local properties. We replace two of them by a second order connection and prove that the holonomity of this connection classifies our materials.