The KITT's initial contribution to the educational activities at Budapest Tech

The Transportation Informatics and Telematics Knowledge Centre (after the abbreviation of its name in Hungarian: ‘KITT’) of Budapest Tech initiated its operation in the Autumn of 2006 thank to the financial support provided by the ‘National Office for Research and Technology’ (NKTH) and the ‘Agency for Research Fund Management and Research Exploitation’ (KPI) using the resources of the ‘Research and Technology Innovation Fund’ within the project No. RET-10/2006. One of the duties of KITT is to give contribution to the educational activities of the host institution of the research consortium, which role is played in this case by Budapest Tech. Regarding education, the main aim is transferring the freshest research results achieved by KITT into the education without considerable delay. Following the first few months of KITT’s activities, Bánki Donát Faculty of Mechanical and Safety Engineering initiated its BSc courses in which, at first time, within the sub-branch of ‘Robot Systems’ of the branch of ‘Mechatronics’, the course ‘Control of Robots’ was launched in the Autumn semester of the academic year 2007/2008. Due to a lucky constellation in the KITT’s Project No. 2.3 entitled ‘Automatic Analysis of Vehicle Behavior’ partly is involved the automatic observation, analysis and control of strongly coupled nonlinear systems of which normally very limited, imprecise, and incomplete ‘a priori’ information is available for the controller, as e.g. in the case of J. K. Tar et al. The KITT’s Initial Contribution to the Educational Activities at Budapest Tech 258 controlling platoons. This situation is quite typical in control of robots, too, for which the study of standard robust approaches as the ‘Robust Sliding Mode / Variable Structure (VS/SM)’ and adaptive techniques as ‘Adaptive Inverse Dynamics (AID)’ or the adaptive algorithm elaborated by Slotine and Li can be designed on the basis of Lyapunov’s 2 Method. In the present paper the part of the curriculum is discussed in which the operation of the VS/SM, the AID, and Slotine’s and Li’s adaptive approaches are compared with that of the novel method elaborated at Budapest Tech (Fixed Point Transformations Based Adaptive Control) by the use of a very simple paradigm. 1 Challenges in Teaching Control of Robots During their education the present students of the BSc course up to this point have obtained various knowledge in a not very ‘integrated’ manner. A part of this knowledge is mainly of lexical nature strictly related to practical applications, the other part is rather devoted to develop a kind of ‘inductive thinking’ also supporting the solution of practical problems, while the most ‘theoretical’ segment is related to Mathematics that scarcely is connected to real applications. The students have some superficial idea of linear operators, linear algebra, matrix product and determinant, the main rules of derivation and integration without having any practice in applying these rules. Similar conditions hold regarding their ideas concerning the relationship between the various models of the reality and the ‘physical reality itself’. For instance, some of them are not aware at all of the fact that several program or software blocks used for quantitative calculations hide or contain certain models of the reality. These students are apt to believe that the numerical values used or provided by the software blocks ‘immediately belong to the reality’ and know nothing of the fact that these quantities are connected to the reality only through certain models. The idea that ‘the same reality’ can be described by various models of various levels of abstraction at a first glance seems to be unbelievable for them. After summarizing the experiences obtained by teaching a whole semester devoted to ‘Kinematics and Dynamics of Robots’, and the first half of the semester ‘Control of Robots’ it cropped up that these subject areas are the first ones in their studies that require a) appropriate skill for abstraction, and b) certain skill for the application of the fundamental mathematical rules in the practice. In spite of the fact that this observation is disappointing and does not shed nice light on the quality of education in the secondary school level institutions, as a teacher, one has to cope with this difficulty and find a way out of the main problem. In the sequel the main point of this idea is briefly reported. Magyar Kutatók 8. Nemzetközi Szimpóziuma 8th International Symposium of Hungarian Researchers on Computational Intelligence and Informatics 259 2 Geometric Way of Thinking as Probably the Best Possible General Solution Considering the historical antecedents of geometric way of thinking in natural sciences makes one persuaded that until the 1 half of the 20 century the development of Mathematics aimed at serving the needs of natural and technical sciences. In the history of the ‘quantitative sciences’ geometric way of thinking always played a pioneering rule. The principles of geometry first were reduced to a small set of axioms by Euclid of Alexandria, a Greek mathematician who worked during the reign of Ptolemy I (323-283 BC) in Egypt. His method of proving mathematical theorems by logical reasoning from accepted first principles remained the backbone of mathematics even in our days, and is responsible for that field's characteristic rigor. Following the pioneering work clarifying the phenomenology of Classical Mechanics by Galilei and Newton, in his fundamental work entitled ‘Mécanique Analytique’ [1] Joseph-Louis Lagrange (1736-1813) solved various optimization problems under constraints, introduced the concept of ‘Reduced Gradient’ and that of what we refer to nowadays as ‘Lagrange Multipliers’. It has to be noted that at that time the concept of ‘linear vector spaces’ was not clarified at all. The first mathematical means of describing quantities with direction, i.e. the ‘quaternions’ introduced by Sir William Rowan Hamilton (1805-1865) appeared not very long time after Lagrange's death [2]. In the 19 century quaternions were generally used for such purposes. For instance, in the first edition of Maxwell's famous ‘Treatise on Electricity and Magnetism’ quaternions were used for describing the ‘directed’ magnetic and electric fields [3]. The first known appearance of what are now called ‘linear algebra’ and the notion of vector spaces is related to Hermann Günther Grassmann (1809-1877), who started to work on the concept from 1832. In 1844, Grassmann published his masterpiece [4] that commonly is referred to as the ‘Theory of Extension’ or ‘Theory of Extensive Magnitudes’. This work was mainly inspired by Lagrange's ‘Mécanique Analytique’. Grassmann showed that once geometry is put into the algebraic form he advocated, then the number three has no privileged role as the number of spatial dimensions: the number of possible dimensions is in fact unbounded. The close relationship between geometry and algebra was realized and strongly utilized by William Kingdon Clifford (1845-1879) who introduced various associative algebras, the so called ‘Clifford Algebras’. As special cases Clifford Algebras contain the algebra of the real, the complex, the dual numbers, the quaternion algebra, and the algebra of octonions (biquaternions) [5]. His Geometric Algebra is widely used in technical sciences as e.g. in computer graphics, robotics, etc. Clifford was the first to suggest that gravitation might be a manifestation of an underlying Riemannian Geometry. J. K. Tar et al. The KITT’s Initial Contribution to the Educational Activities at Budapest Tech 260 Equipped with the concepts of linear vector spaces Marius Sophus Lie (18421899) in his PhD dissertation studied the properties of geometric symmetry transformations [6]. One of his greatest achievements was the discovery that continuous transformation groups (now called after him ‘Lie Groups’) could be better understood by studying the properties of the tangent space of the group elements, that form linear vector spaces (the vector space of the so-called infinitesimal generators), and with the commutator as multiplication also form algebras, the so called ‘Lie Algebras’. In the very fertile period of Mathematics, in the 19 century Georg Friedrich Bernhard Riemann (1826-1866) elaborated the geometry of curved spaces in a special form that made it possible to study physical quantities as tensors even if the geometry of the space differs from the Euclidean Geometry. This concept was very fruitfully used in the General Theory of Relativity. Figure 1 The advantages of geometric way of thinking: it makes it possible to apply lucid and simple problem formulation and argumentation with which we have became familiar already in our childhood on quite ‘abstract’ fields generating the feeling of ‘cozy familiarity’ David Hilbert (1862-1943) extended the concept of the Euclidean Geometry to linear, normed, complete metric spaces in which the norm originates from a scalar product. His invention had extreme advantages in Physics and technical sciences since it makes it possible to apply a way of geometric thinking with which we became familiar from our childhood in the daily experienced Euclidean Geometry of the reality around us (Fig. 1). Stefan Banach (1892-1945) introduced the more general concept, the concept of Banach Spaces, that are linear, normed, complete metric spaces in which the norm not necessarily originates from a scalar product. Magyar Kutatók 8. Nemzetközi Szimpóziuma 8th International Symposium of Hungarian Researchers on Computational Intelligence and Informatics 261 The great practical advantage of Banach's invention is that by adding various norms to the same mathematical set various complete, linear, normed metric spaces can be obtained that offer a wide basis for elaborating diverse practical variants and solutions pertaining to the essentially same basic idea. Vladimir Igorevich Arnold (1937-) studied the Symplectic Geometr