High-precision modeling of CNCs’ spatial errors based on screw theory

A method for establishing machine tool’s spatial error model is put forward based on screw theory, which is different from the traditional error modeling method. By analyzing the position relationship between the ideal coordinate vector and the actual coordinate vector jointly affected by linear errors and angular errors, a single-axis screw conversion matrix error expression is brought up based on screw theory. Meanwhile, the comprehensive spatial error model of the CNC machine tool is derived by considering the influence of the workpiece motion chain and the tool motion chain on the model. Further, to compensating spatial errors of CNCs, such screw theory-based model is embedded in the error compensation system by means of integration of a few specific application examples. And in order to evaluate the compensation effects, an integrated evaluation method of quantitative spatial diagonal calculation and MATLAB simulation is proposed. Application results show that the screw theory-based spatial error model of tool has a very substantial compensation effect, which makes the position error of the machine tool decreased by about 80%.

Inverse Kinematics Data Adaptation to Non-Standard Modular Robotic Arm Consisting of Unique Rotational Modules

The paper describes the original robotic arm designed by our team kinematic design consisting of universal rotational modules (URM). The philosophy of modularity plays quite an important role when it comes to this mechanism since the individual modules will be the building blocks of the entire robotic arm. This is a serial kinematic chain with six degrees of freedom of unlimited rotation. It was modeled in three different environments to obtain the necessary visualizations, data, measurements, structural changes measurements and structural changes. In the environment of the CoppeliaSim Edu, it was constructed mainly to obtain the joints coordinates matching the description of a certain spatial trajectory with an option to test the software potential in future inverse task calculations. In Matlab, the model was constructed to check the mathematical equations in the area of kinematics, the model’s simulations of movements, and to test the numerical calculations of the inverse kinematics. Since the equipment at hand is subject to constant development, its model can also be found in SolidWorks. Thus, the model’s existence in those three environments has enabled us to compare the data and check the models’ structural designs. In Matlab and SolidWorks, we worked with the data imported on joints coordinates, necessitating overcoming certain problems related to calculations of the inverse kinematics. The objective was to compare the results, especially in terms of the position kinematics in Matlab and SolidWorks, provided the initial joint coordinate vector was the same.

Non-Linear Model Dynamics Analysis for Aerospace Engineering Structures Subjected to Non-Steady-State Random Loads

In aerospace engineering, it is customary to employ stochastic analysis methods at the design stage to investigate how the mechanical system responds to random external forces. This is relevant due to high reliability requirements for spacecraft. We developed a method for probabilistic estimation of the dynamic properties of a structure subjected simultaneously to external (additive) and parametric (multiplicative) vibrations. An ordinary non-linear vector differential equation describes the vibrations in the elastic structure. Non-linear position and velocity properties of kinematic pairs may have cusps and discontinuities. We assume that the probabilistic dispersions of respective phase coordinates are close to the normal distribution of probability density. The initial non-linear vibration equations are statistically linearised. The system of differential equations is not rewritten in the canonical form, which means that it is possible to carry out the probabilistic analysis of the system for any external non-steady-state effect. The fundamental matrix of the linearised system is used to find the expected value vector and the correlation function matrix of the phase coordinate vector. The solution consists of a matrix integro-power series containing linear and quadratic terms. Using the method makes it possible to assess the contribution of each external force component to the total result. We consider an example of a non-linear system responding to a stepwise non-steady-state external influence.

Differentiable mapping generated by elliptic parabo­loid complexes

In three-dimensional equiaffine space, we consider a differentiable map generated by complexes with three-parameter families of elliptic paraboloids according to the method proposed by the author in the mate­rials of the international scientific conference on geometry and applica­tions in Bulgaria in 1986, as well as in works published earlier in the sci­entific collection of Differ. Geom. Mnogoobr. Figur. The study is carried out in the canonical frame, the vertex of which coincides with the top of the generating element of the manifold, the first two coordinate vectors are conjugate and lie in the tangent plane of the elliptic paraboloid at its vertex, the third coordinate vector is directed along the main diameter of the generating element so that the ends are, respectively, the sums of the first and third, and also the sums of the second and third coordinate vec­tors lay on a paraboloid, while the indicatrixes of all three coordinate vec­tors describe lines with tangents, parallel to the first coordinate vector. The existence theorem of the mapping under study is proved, according to which it exists and is determined with the arbitrariness of one function of one argument. The systems of equations of the indicatrix and the main directions of the mapping under consideration are obtained. The indicatrix and the cone of the main directions of the indicated mapping are geomet­rically characterized.

Complexes of elliptic cylinders with a characteristic manifold of the generator element in the form of coordinate straight lines

The complex (three-parameter family) of elliptic cylinders is investigated in the three-dimensional affine space, in which the characteristic multiplicity of the forming element consists of three coordinate axes. The focal variety of the forming element of the considered variety is geometrically characterized. Geometric properties of the complex under study were obtained. It is shown that the studied manifold exists and is determined by a completely integrable system of differential equations. It is proved that the focal variety of the forming element of the complex consists of four geometrically characterized points. The center of the ray of the straight-line congruence of the axes of the cylinder, the indicatrix of the second coordinate vector, the second coordinate line and one of the coordinate planes are fixed. The indicatrix of the first coordinate vector describes a one-parameter family of lines with tangents parallel to the second coordinate vector. The end of the first coordinate vector describes a one-parameter family of lines with tangents parallel to the third coordinate vector. The indicatrix of the third coordinate vector and its end describe congruences of planes parallel to the first coordinate plane. The points of the first coordinate line and the first coordinate plane describe one-parameter families of planes parallel to the coordinate plane indicated above.

Integration of knowledge while mathematical courses studying by prospective teachers of Mathematics

The paper focuses on the main directions of integration in modern education. The basis for the examples and conclusions were fundamental studies in the field of Philosophy, Pedagogy and Mathematics. Integration as a general methodological concept in the context of the paper is more concerned with the integration of knowledge in the content of the school course of Mathematics in order to recognize and operate on them in various academic disciplines and the surrounding life. As examples of integration the author gives various concepts from different branches of mathematics (straight line, proportion, symmetry), methods of solving equations and inequalities (functional-graphic, coordinate-vector along with traditional methods), methods of proving identities and inequalities (geometric methods in Algebra and algebraic methods in Geometry). The study of mathematical content in the classroom at school should be a systematic and multifaceted process for the teacher to establish and disclose the links between different concepts, their properties, as well as methods of applying knowledge in a variety of situations. In mathematical courses and the course of Mathematics teaching methods the author offers to pay special attention to prospective teachers ability development to consider mathematical objects from different sides, thereby establishing links between different sections of Mathematics. Various special courses (or elective courses) on relevant subjects are also an important part of prospective teachers professional training, which can be purposefully used to expand inter-subject relations. In addition, the subject of term and final papers can also be made up taking into account different interpretations of mathematical concepts and methods. The author also pays great attention to the organization of such work with students. Only free knowledge enables the teacher to form students solid educational results at the level of knowledge and educational activities for the purpose of their further use in professional activities and in everyday life.

A Variational Derivation of Equations of Motion With Contact Constraints Using SE(3)

For rigid-body systems subjected to non-holonomic constraints, a streamlined method is presented to derive a minimum number of analytical equations of motion. To illustrate the method, a rolling disk problem is considered. In kinematics, an orthonormal coordinate system is attached to the center of mass together with additional coordinate systems introduced to define the connection path. For each coordinate system, a moving frame is defined by explicitly writing the coordinate vector basis and the position vector of the origin, whereby the attitude of the coordinate vector basis and the coordinates of the origin are compactly stored in a 4 × 4 frame connection matrix of the special Euclidean group, SE(3). Contact velocity constraints are transformed to pfaffians to obtain the associated variational constraints. In kinetics, the principle of virtual work is employed. The desired equations of motion are obtained by expressing the translational and angular velocities at the center of mass as the linear functions of the generalized velocities with the coefficients stored in [B]-matrix, and reducing it to [B*]-matrix after incorporating the contact constraints. The method can be easily extended to multi-body systems with both holonomic and non-holonomic constraints.

A new planar triangular element based on the absolute nodal coordinate formulation

In this paper, an improved three-node incomplete cubic planar triangular element is proposed based on the two recently reported absolute nodal coordinate formulation triangular elements. Compared with the existing absolute nodal coordinate formulation elements, a different set of polynomial basis is used to develop the new element using the method analogous to the one used in the conventional Zienkiewicz triangular element. Concise shape functions are obtained by employing both Cartesian and area coordinate sets and the concept of independent area gradient coordinate vector. From the view of the order of the polynomial basis, the criterion for developing incomplete cubic absolute nodal coordinate formulation triangular element that captures the quadratic accuracy is presented. Additionally, the algebraic constraint method used in developing the incomplete cubic triangular element is discussed. Based on the criterion, the proposed element is compared analytically with the previous incomplete cubic element. On the other hand, the proposed element is evaluated using both the static and dynamic numerical examples. The element successfully passes the patch test. The results obtained by the proposed element in this paper agree well with analytical solutions or those given by the full cubic element/general commercial finite element software. The higher accuracy, better convergence of the proposed element and the criterion are verified.

POTENTIALS OF A FROBENIUS-LIKE STRUCTURE

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame, which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.