Search Results (6)

This paper presents the design and implementation of a Model-based Predictive Control (MPC) strategy integrated within a modular multilayer architecture for a three-wheeled omnidirectional mobile robot, the Robotino 4 from Festo. The implemented architecture is organized into three hierarchical layers to support modularity and system scalability. The upper layer is responsible for trajectory planning. This planned trajectory is forwarded to the intermediate layer, where the MPC computes the optimal velocity commands to follow the reference path, taking into account the kinematic model and actuator constraints of the robot. Finally, these velocity commands are processed by the lower layer, which uses three independent PID controllers to regulate the individual wheel speeds. To evaluate the proposed control scheme, it was implemented in MATLAB R2024a using a lemniscate trajectory as the reference. The MPC problem was formulated as a quadratic optimization problem that considered the three states: the global position coordinates and orientation angle. The simulation included state estimation errors and motor dynamics, which were experimentally identified to closely match real-world behavior. The simulation and experimental results demonstrate the capability of the MPC to track the lemniscate trajectory efficiently. Notably, the close agreement between the simulated and experimental results validated the fidelity of the simulation model. In a real-world scenario, the MPC controller enabled simultaneous regulation of both the position and orientation, which offered a greater performance compared with approaches that assume a constant orientation. Full article

In this paper, the very fundamental geometrical characteristics of the Mylar balloon like the profile curve, height, volume, arclength, surface area, crimping factor, etc. are recognized as geometrical moments ${\mathcal{I}}_n\left(x\right)$ and ${\mathcal{I}}_n$ and this observation has been used to introduce an infinite family of surfaces ${\mathcal{S}}_n$ specified by the natural numbers $n=0,1,2,\dots$. These surfaces are presented via explicit formulas (through the incomplete Euler’s beta function) and can be identified as an interesting family of balloons. Their parameterizations is achieved relying on the well-known relationships among elliptic integrals, beta and gamma functions. The final results are expressed via the fundamental mathematical constants, such as $\pi$ and the lemniscate constant $\varpi$. Quite interesting formulas for recursive calculations of various quantities related to associated figures modulo four are derived. The most principal results are summarized in a table, illustrated via a few graphics, and some direct relationships with other fundamental areas in mathematics, physics, and geometry are pointed out. Full article

In this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemniscate constants $A,B$ and Gauss’s constant G, which are used to compare with the classical pursuit curve. The analogous spherical geometry problem is solved using a spherical curve defined by the Gudermannian function. It is shown that the results agree with the angle-preserving property of Mercator and Stereographic projections. The Mercator and Stereographic projections also reveal the symmetry of this curve with respect to Spherical and Logarithmic Spirals. The geometric properties of the spherical curve are proved in two ways, analytically and using a lemma about spherical angles. A similar lemma for the planar case is also mentioned. The paper shows symmetry/asymmetry between the spherical and planar cases and the derivation of properties of these curves as limiting cases of some plane and spherical geometry results. Full article

Starting with identifications of the very fundamental geometric characteristics of a Mylar balloon such as the profile curve, height, volume, arclength, surface area, crimping factor, etc., using the geometrical moments $I_n\left(x\right)$ and $I_n$, we present explicit formulas for them and those of the mechanical moments of both solid and hollow balloons of arbitrary order. This is achieved by relying on the recursive relationships among elliptic integrals and the final results are expressed via the fundamental mathematical constants such as $\pi$, lemniscate constant $\tilde{\omega }$, and Gauss’s constant G. An interesting periodicity modulo 4 was detected and accounted for in the final formulas for the moments. The principal results are illustrated by two tables, a few graphics, and some direct relationships with other fundamental areas in mathematics, physics and geometry are pointed out. Full article

In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition $\Re \left(\frac{f\left(z\right)}{g\left(z\right)}\right)>0$, for some analytic function, g, with $\Re \left(\frac{{\left(z+1\right)}^{-2n}}{z}\right)g\left(z\right)>0,$∀$n\in \mathbb{N}.$ We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp. Full article

We present surprisingly simple closed-form solutions for electric fields and electric potentials at arbitrary position $\left(x,y\right)$ within a plane crossed by infinitely long line charges at regularly repeating positions using angular or elliptic functions with complex arguments. The lattice sums for the electric-field components and the electric potentials could be exactly solved, and the duality symmetry of trigonometric and lemniscate functions occurred in some solutions. The results may have relevance in calculating field configurations with rectangular boundary conditions. Several series related to Gauss’s constant are presented, established either as corollary results or via parallel investigations conducted in the spirit of experimental mathematics. Full article