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This article is concerned with developing an intelligent system for the control of a wheeled robot. An algorithm for training an artificial neural network for path planning is proposed. The trajectory ensures steering optimal motion from the current position of the mobile robot to a prescribed position taking its orientation into account. The proposed control system consists of two artificial neural networks. One of them serves to specify the position and the size of the obstacle, and the other forms a continuous trajectory to reach it, taking into account the information received, the coordinates, and the orientation at the point of destination. The neural network is trained on the basis of samples obtained by modeling the equations of motion of the wheeled robot which ensure its motion along trajectories in the form of Euler’s elastica.

We relate the sub-Riemannian geometry on the group of rigid motions of the plane to 'bicycling mathematics'. We show that this geometry's geodesics correspond to bike paths whose front tracks are either non-inflectional Euler elasticae or straight lines, and that its infinite minimizing geodesics (or 'metric lines') correspond to bike paths whose front tracks are either straight lines or 'Euler's solitons' (also known as syntractrix or convicts' curves). [ABSTRACT FROM AUTHOR]

This article is concerned with developing an intelligent system for the control of a wheeled robot. An algorithm for training an artificial neural network for path planning is proposed. The trajectory ensures steering optimal motion from the current position of the mobile robot to a prescribed position taking its orientation into account. The proposed control system consists of two artificial neural networks. One of them serves to specify the position and the size of the obstacle, and the other forms a continuous trajectory to reach it, taking into account the information received, the coordinates, and the orientation at the point of destination. The neural network is trained on the basis of samples obtained by modeling the equations of motion of the wheeled robot which ensure its motion along trajectories in the form of Euler's elastica. [ABSTRACT FROM AUTHOR]

The paper is devoted to multiple solutions of the classical problem on stationary configurations of an elastic rod on a plane; we describe boundary values for which there are more than two optimal configurations of a rod (optimal elasticae). We define sets of points where three or four optimal elasticae come together with the same value of elastic energy. We study all configurations that can be translated into each other by symmetries, i.e., reflections at the center of the elastica chord and reflections at the middle perpendicular to the elastica chord. For the first symmetry, the ends of the rod are directed in opposite directions, and the corresponding boundary values lie on a disk. For the second symmetry, the boundary values lie on a Möbius strip. As a result, we study both sets numerically and in some cases analytically; in each case, we find sets of points with several optimal configurations of the rod. These points form the currently known part of the reachability set where elasticae lose global optimality. [ABSTRACT FROM AUTHOR]

We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on a step 3 Carnot group of dimension 5. We prove the conjectured cut times of Yu. Sachkov for the sub-Riemannian Cartan problem. Along the proof, we obtain a comparison with the known cut times in the sub-Riemannian Engel group, and a sufficient (generic) condition for the uniqueness of the length minimizer between two points. Hence we reduce the optimal synthesis to solving a certain system of equations in elliptic functions. [ABSTRACT FROM AUTHOR]

The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic rank two sub-Riemannian problems on four-dimensional manifolds. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping. [ABSTRACT FROM AUTHOR]

The article discusses the technology for automated neural network monitoring of the vineyard's physiological condition. Images of leaves, obtained using an unmanned aerial vehicle (UAV), are the main indicator of the physiological vineyard's condition. The proposed solution is based on the integrated use of convolutional neural network method (CNN) and machine vision technologies. To determine the optimal neural network (NN) model, a variant analysis was carried out. In accordance with its results, the YOLOv7 model was chosen, which satisfies the introduced time limit and provides the required detection quality. The training of the YOLOv7 neural network was implemented in the Python environment using the PyTorch framework and the OpenCV computer vision library. The dataset consisting of 6320 images of grape leaves (including healthy and diseased ones) has been used for neural network training. The obtained results showed that the detection accuracy is at least 91%. Visualization of monitoring results has been carried out using heatmap, allowing to obtain information about vineyard physiological condition in dynamics. The proposed mathematical model allows to calculate the monitored vineyard's area made by one complex per day. The obtained results showed that effective monitoring area using one DJI Phantom 4 UAV per day is 2.5 hectares. [ABSTRACT FROM AUTHOR]

Euler's elastica is defined by a critical point of the total squared curvature under the fixed length constraint, and its L p -counterpart is called p-elastica. In this paper we completely classify all p-elasticae in the plane and obtain their explicit formulae as well as optimal regularity. To this end we introduce new types of p-elliptic functions which streamline the whole argument and result. As an application we also classify all closed planar p-elasticae. [ABSTRACT FROM AUTHOR]

Control technology for wheeled mobile robots is one of the core focuses in the current field of robotics research. Within this domain, model-free adaptive control (MFAC) methods, with their advanced data-driven strategies, have garnered widespread attention. The unique characteristic of these methods is their ability to operate without relying on prior model information of the control system, which showcases their exceptional capability in ensuring closed-loop system stability. This paper extensively details three dynamic linearization techniques of MFAC: compact form dynamic linearization, partial form dynamic linearization and full form dynamic linearization. These techniques lay a solid theoretical foundation for MFAC. Subsequently, the article delves into some advanced MFAC schemes, such as dynamic event-triggered MFAC and iterative learning MFAC. These schemes further enhance the efficiency and intelligence level of control systems. In the concluding section, the paper briefly discusses the future development potential and possible research directions of MFAC, aiming to offer references and insights for future innovations in control technology for wheeled mobile robots. [ABSTRACT FROM AUTHOR]

In this work we study extremals on Lie groups G endowed with a left invariant polyhedral Finsler structure. We use the Pontryagin’s Maximal Principle (PMP) to find curves on the cotangent bundle of the group, such that its projections on G are extremals. Let g and g ∗ be the Lie algebra of G and its dual space respectively. We represent this problem as a control system a ′ (t) = - ad ∗ (u (t)) (a (t)) of Euler-Arnold type equation, where u (t) is a measurable control in the unit sphere of g and a (t) is an absolutely continuous curve in g ∗ . A solution (u (t) , a (t)) of this control system is a Pontryagin extremal and a (t) is its vertical part. In this work we show that for a fixed vertical part of the Pontryagin extremal a (t) , the uniqueness of u (t) such that (u (t) , a (t)) is a Pontryagin extremal can be studied through an asymptotic curvature of a (t) . [ABSTRACT FROM AUTHOR]

An extended four-dimensional version of the traditional Petitot–Citti–Sarti model on contour completion in the visual cortex is examined. The neural configuration space is considered as the group of similarity transformations, denoted as M = SIM(2) . The left-invariant subbundle of the tangent bundle models possible directions for establishing neural communication. The sub-Riemannian distance is proportional to the energy expended in interneuron activation between two excited border neurons. According to the model, the damaged image contours are restored via sub-Riemannian geodesics in the space M of positions, orientations and thicknesses (scales). We study the geodesic problem in M using geometric control theory techniques. We prove the existence of a minimal geodesic between arbitrary specified boundary conditions. We apply the Pontryagin maximum principle and derive the geodesic equations. In the special cases, we find explicit solutions. In the general case, we provide a qualitative analysis. Finally, we support our model with a simulation of the association field. [ABSTRACT FROM AUTHOR]

Starting from the programmable motion performance of modern actuation technology, this study elucidates that the degree of quick-return in a mechanical system is no longer entirely dependent on mechanism design. A theoretical framework is proposed for analyzing and synthesizing the degree of quick-return in contemporary mechanical systems based on time efficiency. This study adopts a functional perspective, first extending the traditional definition describing the degree of quick-return mechanism to encompass broader mechanical systems. Subsequently, we present a general formula for calculating the time-ratio and travel velocity-ratio coefficients under contemporary technological conditions, supported by a detailed mathematical derivation. Finally, a corresponding procedure is established for analyzing and synthesizing the degree of quick-return in a mechanical system, fostering improved collaboration between managers and designers. Case studies demonstrate that designers can achieve the desired degree of quick-return in mechanical systems and robots by leveraging time efficiency. The theoretical framework introduced in this paper is versatile and can benefit various industries, notably the field of robotics. This broadens the applicability of the degree of quick-return concept, marking a significant advancement in enhancing the time efficiency of mechanical systems. [ABSTRACT FROM AUTHOR]

The object of the research is the application of deep learning algorithms using an improved mathematical lossless image compression method for recognizing and identifying dead trees in aerospace images. The main problem that has been solved is the archiving of images due to their large volume on disk and the possibility of their further processing by deep learning methods such as convolutional and capsule neural networks, which have shown high efficiency and accuracy in image recognition and classification tasks using the proposed new image compression method. The article presents a comparative analysis of the performance of three YOLO (You Only Look Once) models with different types of architectures, such as YOLOv5, YOLOv7 and YOLOv8, to assess the effectiveness of their work for the task of recognizing aerospace tree images obtained from satellites, drones, and aircrafts. Comprehensive analysis of YOLO models presents that model YOLO v8 turned out to be most effective with a positive accuracy of 88.2 %, a recall of 77.4 %, and a mAP50 score of 87.2 %. Moreover, the average detection time was only 0.052 seconds for each image, even though the model size remains very small - 21.5 MB. These results suggest a much better usage of time and precise identification of dead trees, and classified targets with high efficiency. From the research, there is significant prospects of global forest management especially on forest reduction and protection of ecosystems through accurate assessment on the health of forestry. The proposed approach is universal and can be used in real life conditions, providing a good compromise of the speed, accuracy and resources required for forest monitoring and management. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence theorem for longest paths in sub-Lorentzian problems, which generalizes the classical theorem for globally hyperbolic Lorentzian manifolds. We specifically address the case of invariant structures on homogeneous spaces, as the conditions for the existence theorem in this case can be significantly simplified. In particular, it turns out that longest paths exist for any left-invariant sub-Lorentzian structures on Carnot groups. [ABSTRACT FROM AUTHOR]

Precision control of multiple robotic fish visual navigation in complex underwater environments has long been a challenging issue in the field of underwater robotics. To address this problem, this paper proposes a multi-robot fish obstacle traversal technique based on the combination of cross-modal variational autoencoder (CM-VAE) and imitation learning. Firstly, the overall framework of the robotic fish control system is introduced, where the first-person view of the robotic fish is encoded into a low-dimensional latent space using CM-VAE, and then different latent features in the space are mapped to the velocity commands of the robotic fish through imitation learning. Finally, to validate the effectiveness of the proposed method, experiments are conducted on linear, S-shaped, and circular gate frame trajectories with both single and multiple robotic fish. Analysis reveals that the visual navigation method proposed in this paper can stably traverse various types of gate frame trajectories. Compared to end-to-end learning and purely unsupervised image reconstruction, the proposed control strategy demonstrates superior performance, offering a new solution for the intelligent navigation of robotic fish in complex environments. [ABSTRACT FROM AUTHOR]

The article deals with the development and determination of the uncertainty of a calibration tool for tilt sensors for the control of motor vehicle instability. The stability of motor vehicles is critical, especially when driving on inclined terrain, when instability and overturning of the vehicle can occur. For this purpose, it is necessary to install and calibrate a suitable tilt sensor. [ABSTRACT FROM AUTHOR]

Automated guided vehicles (AGVs) have become prevalent over the last decade. However, numerous challenges remain, including path planning, security, and the capacity to operate safely in unstructured environments. This study proposes an obstacle avoidance system that leverages deep action learning (DAL) to address these challenges and meet the requirements of Industry 4.0 for AGVs, such as speed, accuracy, and robustness. In the proposed approach, the DAL is integrated into an AGV platform to enhance its visual navigation, object recognition, localization, and decision-making capabilities. Then DAL itself was introduced to combine the work of You Only Look Once (YOLOv4), speeded-up robust features (SURF), and k-nearest neighbor (kNN) and AGV control in indoor visual navigation. The DAL system triggers SURF to differentiate two navigation images, and kNN is used to verify visual distance in real time to avoid obstacles on the floor while searching for the home position. The testing findings show that the suggested system is reliable and fits the needs of advanced AGV operations. [ABSTRACT FROM AUTHOR]

The authors have considered the gas-dynamic influence upon the mechanical properties, chemical composition, microhardness, and geometry of single-rim welds made of 30ХГСА steel when welding with a consumable electrode under a double-jet gas shield. Their regressional relationships on the selected controlled welding parameters have been developed. It has been found that the gas-dynamic effect of a dynamic shield gas jet has a controlling influence on the formation of welds made of alloy-treated 30ХГСА steel. [ABSTRACT FROM AUTHOR]

Nonlinear control-affine systems described by ordinary differential equations with time-varying vector fields are considered in the paper. We propose a unified control design scheme with oscillating inputs for solving the trajectory tracking and stabilization problems under the bracket-generating condition. This methodology is based on the approximation of a gradient-like dynamics by trajectories of the designed closed-loop system. As an intermediate outcome, we characterize the asymptotic behavior of solutions of the considered class of nonlinear control systems with oscillating inputs under rather general assumptions on the generating potential function. These results are applied to examples of nonholonomic trajectory tracking and obstacle avoidance. [ABSTRACT FROM AUTHOR]

A three-dimensional reachable set for a nonlinear controlled object "Dubins car" is investigated. The control is the angular velocity of rotation of the linear velocity vector. An integral quadratic constraint is imposed on the control. Based on the Pontryagin maximum principle, a description of the motions generating the boundary of the reachable set is given. The motions leading to the boundary are optimal Euler elasticae. Simulation results are presented. [ABSTRACT FROM AUTHOR]

Tractors with multiple trailers are widely applied means of transport in manufacturing systems. There exist numerous designs of trailers and tractors, making the estimation of the system trajectory and the required transportation corridor a complex task. It is also difficult to achieve the same trajectory for a manually operated tractor for multiple runs. The problem is complicated if there are multiple towed trailers or a dynamic drive on slippery ground. One approach is to replace the driver with an automated steering system. This paper presents a dynamic model of a tractor with multiple trailer system, based on the Lagrange formalism, which is controlled by artificial neural networks. To account for the slip phenomenon, a sigmoidal tire model was used. The algorithm of the artificial neural network provides the most appropriate input parameters for tractor steering for a given transportation area. The input parameters are the torques applied to the tractor wheels and are determined by the algorithm based on the data collected by the LiDAR scanner during the train run. These data include distances for each unit from the obstacle (e.g., wall), information about the occurrence of a collision, and the distance traveled by the tractor. The simulation results of the integration of the dynamic model and the neural network modeled are presented in a graphic form. The proposed algorithm ensures a collision-free ride of the system. [ABSTRACT FROM AUTHOR]

A map construction method based on a collision probability model and an improved A* algorithm is proposed to address the issues of insufficient security in mobile robot map construction and path planning in complex environments. The method is based on modeling the asymmetry of paths, which complicates problem solving. Firstly, this article constructs a collision probability function model, and based on this model it is fused with the obstacle grid map, which is based on the grid method, to draw a collision probability grid map (CPGM) containing collision probability information. Secondly, incorporating the collision probability values from the CPGM into the actual cost function of the traditional A* algorithm improves the security of path planning in complex environments. The experimental results show that the improved A* algorithm decreases the percentage of dangerous nodes in complex environments by 69.23%, shortens the path planning length by 19.52%, reduces the search time by 16.8%, and reduces the number of turns by 46.67%. Therefore, the method in this paper solves the problem of traditional grid maps lacking security information and can plan a path with higher security and which is smoother, improving the security and robustness of mobile robot autonomous navigation in complex environments. [ABSTRACT FROM AUTHOR]

The rise in Unmanned Aerial Vehicle (UAV) utilization in science, research, and measurement procedures offers an advancement in various diagnostics methods and an improvement in measurement accuracy and versatility. This also applies to large Antenna Systems (AS), in FM and DVB-T frequency bands, parameter measurements, as well as to their diagnostics. The article presents the electromagnetic field intensity measurements performed by an autonomous quadcopter at a sufficient distance from the AS along with a brief description of the measurement procedure. The measured data are used for faulty antenna identification in a large AS. The article follows upon previous study regarding a large AS diagnostics and shows a comparison between the data calculated by a computation software and the data measured with the use of a quadcopter and specialized measurement device. Based on the results of the comparison, the individual faulty antenna is identified. The quadcopter measurement can be performed during continuous operation of the transmitter (AS is radiating). [ABSTRACT FROM AUTHOR]

We study the geometric structures associated with curvature radii of curves with values on a Riemannian manifold (M, g). We show the existence of sub-Riemannian manifolds naturally associated with the curvature radii and we investigate their properties. In the particular case of surfaces these sub-Riemannian structures are of Engel type. The main character of our construction is a pair of global vector fields f 1 , f 2 , which encodes intrinsic information on the geometry of (M, g). [ABSTRACT FROM AUTHOR]

We study the local geometry of the sub-Finsler structure induced by a sub-Riemannian metric on a 3-dimensional manifold. We provide a description of the upper part of the cut locus of short geodesics, in some non generic cases. [ABSTRACT FROM AUTHOR]

We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in SE 2 and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part. [ABSTRACT FROM AUTHOR]

Dispatching and cooperative trajectory planning for multiple autonomous forklifts in a warehouse is a widely applied research topic. The conventional methods in this domain regard dispatching and planning as isolated procedures, which render the overall motion quality of the forklift team imperfect. The dispatching and planning problems should be considered simultaneously to achieve optimal cooperative trajectories. However, this approach renders a large-scale nonconvex problem, which is extremely difficult to solve in real time. A joint dispatching and planning method is proposed to balance solution quality and speed. The proposed method is characterized by its fast runtime, light computational burden, and high solution quality. In particular, the candidate goals of each forklift are enumerated. Each candidate dispatch solution is measured after concrete trajectories are generated via an improved hybrid A* search algorithm, which is incorporated with an artificial neural network to improve the cost evaluation process. The proposed joint dispatching and planning method is computationally cheap, kinematically feasible, avoids collisions with obstacles/forklifts, and finds the global optimum quickly. The presented motion planning strategy demonstrates that the integration of a neural network with the dispatching approach leads to a warehouse filling/emptying mission completion time that is 2% shorter than the most efficient strategy lacking machine-learning integration. Notably, the mission completion times across these strategies vary by approximately 15%. [ABSTRACT FROM AUTHOR]

Artificial neural networks can solve various tasks in computer vision, such as image classification, object detection, and general recognition. Our comparative study deals with four types of artificial neural networks—multilayer perceptrons, probabilistic neural networks, radial basis function neural networks, and convolutional neural networks—and investigates their ability to classify 2D matrix codes (Data Matrix codes, QR codes, and Aztec codes) as well as their rotation. The paper presents the basic building blocks of these artificial neural networks and their architecture and compares the classification accuracy of 2D matrix codes under different configurations of these neural networks. A dataset of 3000 synthetic code samples was used to train and test the neural networks. When the neural networks were trained on the full dataset, the convolutional neural network showed its superiority, followed by the RBF neural network and the multilayer perceptron. [ABSTRACT FROM AUTHOR]

An improved butterfly optimization algorithm (IBOA) is proposed to overcome the disadvantages, including slow convergence, generation of local optimum solutions, and deadlock phenomenon, of the optimization algorithm in the path planning of mobile robots. A path-planning grid model is established based on an improved obstacle model. First, the population diversity is improved by introducing kent mapping during population position renewal in the normal butterfly optimization algorithm (BOA) to enhance the global search ability of the butterfly population. Second, an adaptive weight coefficient is introduced in the renewal process of each generation to increase the convergence speed and accuracy. An opposition-based learning strategy based on convex lens imaging is introduced to help the butterfly population jump out of the local optimum. Finally, a mutation strategy is introduced to solve the path planning problem. On this basis, two path simplification strategies are proposed to make up for the shortcomings of planning paths in grid maps. The shortest path lengths solved by IBOA, BOA, and GA in the 20 × 20 map are 30.97, 31.799, and 31.799, respectively. The numbers of iterations for the shortest paths searched by IBOA, BOA, and GA are 14, 24, and 38 in that order. The shortest path lengths solved by IBOA, BOA and GA in the 40 × 40 map are 63.84, 65.60, and 65.84, respectively. The number of iterations for the shortest paths searched by IBOA, BOA and GA are 32, 40, and 46, respectively. Simulation results show that IBOA has a strong ability to solve robot path planning problems and that the proposed path simplification strategy can effectively reduce the length of the optimal path in the grid map to solve the path planning problem of mobile robots. The shortest paths solved by IBOA in 20 × 20 and 40 × 40 maps are simplified to lengths of 30.2914 and 61.03, respectively. [ABSTRACT FROM AUTHOR]

In mathematical control theory, a Dubins car is a nonlinear motion model described by differential relations, in which the scalar control determines the instantaneous angular rate of rotation. The value of the linear velocity is assumed to be constant. The phase vector of the system is three-dimensional. It includes two coordinates of the geometric position and one coordinate having the meaning of the angle of inclination of the velocity vector. This model is popular and is used in various control tasks related to the motion of an aircraft in a horizontal plane, with a simplified description of the motion of a car, small surface and underwater vehicles, etc. Scalar control can be constrained either by a symmetric constraint (when the minimum rotation radii to the left and right are the same) or asymmetric constraint (when rotation is possible in both directions, but the minimum rotation radii are not the same). Usually, problems with symmetric and asymmetric constraints are considered separately. It is shown that when constructing the reachability set at the moment, the case of an asymmetric constraint can be reduced to a symmetric case. [ABSTRACT FROM AUTHOR]

Abnormal trajectories are of particular interest for sub-Riemannian geometry, because the most complicated singularities of the sub-Riemannian metric are located just near such trajectories. Important open questions in sub-Riemannian geometry are to establish whether the abnormal length minimizers are smooth and to describe the set filled with abnormal trajectories starting from a fixed point. For example, the Sard conjecture in sub-Riemannian geometry states that this set has measure zero. In this paper, we consider this and other related properties of such a set for the left-invariant sub-Riemannian problem with growth vector . We also study the global and local optimality of abnormal trajectories and obtain their explicit parametrization. [ABSTRACT FROM AUTHOR]

A Finsler analog of the Lobachevsky plane is the Lie group of proper affine transformations of the real line with a left-invariant Finsler structure generated by a convex compact set in the Lie algebra with the origin in its interior. We consider the isoperimetric problem on this Lie group, with the volume form also taken to be left-invariant. This problem is formulated as an optimal control problem. Applying the Pontryagin maximum principle, we find the optimal isoperimetric loops in an explicit form in terms of convex trigonometry functions. We also present a generalized isoperimetric inequality in a parametric form. [ABSTRACT FROM AUTHOR]

We consider a time-optimal problem for a car model that can move forward on a plane and turn with a given minimum turning radius. Trajectories of this system are applicable in image processing for the detection of salient lines. We prove the controllability and existence of optimal trajectories. Applying the necessary optimality condition given by the Pontryagin maximum principle, we derive a Hamiltonian system for the extremals. We provide qualitative analysis of the Hamiltonian system and obtain explicit expressions for the extremal controls and trajectories. [ABSTRACT FROM AUTHOR]