Search Results (156)

Trajectory planning aims to compute an optimal path and velocity of an agent through the minimization of a cost function. This paper proposes a just-in-time routing method, incorporating the stochastic minimization of a cost function, which ingests the effect of the agent’s environment evolving in space and time. The environment is considered known at present, but the uncertainty increases when advancing in time. To compute the optimal routing in such an uncertain environment, Euler–Lagrange equations will be formulated in a stochastic setting, to obtain a probabilistic optimal planning. With the cost function approximated by using a surrogate modeling based on deep neural networks, a neural formulation of the stochastic Euler–Lagrange equations is proposed and employed. Full article

This paper examines the impact of symmetry and asymmetry on the dynamic modeling and nonlinear control of a mobile robot with Ackermann steering geometry. A neural network-based residual model is incorporated as a novel control enhancement. This study presents a control-oriented formulation that addresses both idealized symmetric dynamics and real-world asymmetric behaviors caused by actuator imperfections, tire slip, and environmental variability. Using the Euler–Lagrange formalism, the robot’s dynamic equations are derived, and a modular simulation framework is implemented in MATLAB/Simulink R2022a, that incorporates distinct steering and propulsion subsystems. Symmetric elements, such as the structure of the inertia matrix and kinematic constraints, are contrasted with asymmetries introduced through actuator lag, unequal tire stiffness, and nonlinear friction. A residual neural network term is introduced to capture unmodeled dynamics and improve the robustness. The simulation results show that the control strategy, originally developed under symmetric assumptions, remains effective when adapted to systems exhibiting asymmetry, such as actuator delays and tire slip. Explicitly modeling these asymmetries enhances the precision of trajectory tracking and the overall system robustness, particularly in scenarios involving varied terrain and obstacle-rich environments. Full article

Fractional calculus of variations for a broad class of fractional operators with a general analytic kernel function is considered. Using techniques from variational analysis, we derive first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the Weierstrass necessary condition, the Legendre condition, and finally the Legendre–Clebsch condition. Our results are new in the sense that the Euler–Lagrange equation is based on duality theory, and thus build up only with left fractional operators. The Weierstrass necessary condition is a variant of strong necessary optimality condition, and it is derived from maximum condition of Pontryagin for this general analytic kernels. The Legendre–Clebsch condition is obtained under normality assumptions on data because of equality constraints. Full article

The swinging sticks pendulum is an intriguing physical system that exemplifies the intersection of Lagrangian mechanics and chaos theory. It consists of a series of slender, interconnected metal rods, each with a counterweighted end that introduces an asymmetrical mass distribution. The rods are arranged to pivot freely about their attachment points, enabling both rotational and translational motion. Unlike a simple pendulum, this system exhibits complex and chaotic behavior due to the interplay between its degrees of freedom. The Lagrangian formalism provides a robust framework for modeling the system’s dynamics, incorporating both rotational and translational components. The equations of motion are derived from the Euler–Lagrange equations and lack closed-form analytical solutions, necessitating the use of numerical methods. In this work, we employ the Bulirsch–Stoer method, a high-accuracy extrapolation technique based on the modified midpoint method, to solve the equations numerically. The system possesses four fixed points, each one associated with a different level of energy. The fixed point with the lowest energy level is a center, around which small perturbations are studied. The other three fixed points are unstable. The maximum energy used for the perturbations is $0.001\%$ larger than the lowest equilibrium energy. When the system’s total energy is low, nonlinear terms in the equations can be neglected, allowing for a linearized treatment based on small-angle approximations. Under these conditions, the pendulum oscillates with small amplitudes around a stable equilibrium point. The resulting motion is analyzed using tools from nonlinear dynamics and Fourier analysis. Several trajectories are generated and examined to reveal frequency interactions and the emergence of complex dynamical behavior. When a small initial perturbation is applied to one rod, its motion is characterized by a single frequency with significantly greater amplitude and angular velocity compared to the second rod. In contrast, the second rod displayed dynamics that involved two frequencies. The present study, to the best of our knowledge, is the first attempt to describe the dynamical behavior of this pendulum. Full article

The optical flow problem and image registration problem are treated as optimal control problems associated with Fokker–Planck equations with controller u in the drift term. The payoff is of the form ${\displaystyle \frac{1}{2}}|y\left(T\right)-y_1{|}^2+\alpha {\int }_0^T{|u\left(t\right)|}_4^{4}dt$, where $y_1$ is the observed final state and $y=y^u$ is the solution to the state control system. Here, we prove the existence of a solution and obtain also the Euler–Lagrange optimality conditions which generate a gradient type algorithm for the above optimal control problem. A conceptual algorithm to compute the approximating optimal control and numerical implementation of this algorithm is discussed. Full article

System representation via computational graphs has become a cornerstone of modern machine learning, underpinning the gradient-based training of complex models. We have previously introduced the Partial Lagrangian Method—a divide-and-conquer approach that decomposes the Lagrangian into link-wise components—to derive and evaluate the equations of motion for robot systems with dynamically changing structures. That method leverages the symbolic expressiveness of computational graphs with automatic differentiation to streamline dynamic analysis. In this paper, we advance this framework by establishing a principled way to encode time-dependent differential equations as computational graphs. Our approach, which augments the state vector and applies the chain rule, constructs fully time-independent graphs directly from the Lagrangian, eliminating the erroneous time-derivative embeddings that previously required manual correction. Because our transformation is derived from first principles, it guarantees graph correctness and generalizes to any system governed by variational dynamics. We validate the method on a simple serial-link robotic arm, showing that it faithfully reproduces the standard equations of motion without graph failure. Furthermore, by compactly representing state variables, the resulting computational graph achieves a seven-fold reduction in evaluation time compared to our prior implementation. The proposed framework thus offers a more intuitive, scalable, and efficient design and analysis of complex dynamic systems. Full article

A geodesic or geodetic line on a sphere is called the orthodrome. Research has shown that the orthodrome can be defined in a large number of ways. This article provides an overview of various definitions of the orthodrome. We recall the definitions of the orthodrome according to the greats of geodesy, such as Bessel and Helmert. We derive the equation of the orthodrome in the geographic coordinate system and in the Cartesian spatial coordinate system. A geodesic on a surface is a curve for which the geodetic curvature is zero at every point. Equivalent expressions of this statement are that at every point of this curve, the principal normal vector is collinear with the normal to the surface, i.e., it is a curve whose binormal at every point is perpendicular to the normal to the surface, and that it is a curve whose osculation plane contains the normal to the surface at every point. In this case, the well-known Clairaut equation of the geodesic in geodesy appears naturally. It is found that this equation can be written in several different forms. Although differential equations for geodesics can be found in the literature, they are solved in this article, first, by taking the sphere as a special case of any surface, and then as a special case of a surface of rotation. At the end of this article, we apply calculus of variations to determine the equation of the orthodrome on the sphere, first in the Bessel way, and then by applying the Euler–Lagrange equation. Overall, this paper elaborates a dozen different approaches to orthodrome definitions. Full article

This study explores the dynamics of particle motion in pseudo-Galilean $3-$space $G_3^1$ by considering actions that incorporate both curvature and torsion of trajectories. We consider a general energy functional and formulate Euler–Lagrange equations corresponding to this functional under some boundary conditions in $G_3^1$. By adapting the geometric tools of the Frenet frame to this setting, we analyze the resulting variational equations and provide illustrative solutions that highlight their structural properties. In particular, we examine examples derived from natural Hamiltonian trajectories in $G_3^1$ and extend them to reflect the distinctive geometric features of pseudo-Galilean spaces, offering insight into their foundational behavior and theoretical implications. Full article

Integer-order models cannot characterize the dynamic behavior of the flexible two-link manipulator (FTLM) system accurately due to its viscoelastic characteristics and flexible oscillation. Hence, this paper proposes a fractional-order modeling method and identification algorithm for the FTLM system. Firstly, we exploit the memory and history-dependent properties of fractional calculus to describe the flexible link’s viscoelastic potential energy and viscous friction. Secondly, we establish a fractional-order differential equation for the flexible link based on the fractional-order Euler–Lagrange equation to characterize the flexible oscillation process accurately. Accordingly, we derive the fractional-order model of the FTLM system by analyzing the motor–link coupling as well as the symmetry of the system structure. Additionally, a system identification algorithm based on the multi-innovation integration operational matrix (MIOM) is proposed. The multi-innovation technique is combined with the least-squares algorithm to solve the operational matrix and achieve accurate system identification. Finally, experiments based on actual data are conducted to verify the effectiveness of the proposed modeling method and identification algorithm. The results show that the MIOM algorithm can improve system identification accuracy and that the fractional-order model can describe the dynamic behavior of the FTLM system more accurately than the integer-order model. Full article

The mechanics of an elastic sheet reinforced with fiber mesh is investigated when undergoing bilateral in-plane bending and stretching. The strain energy of FRC is formulated by accounting for the matrix strain energy contribution and the fiber network deformations of extension, flexure, and torsion, where the strain energy potential of the matrix material is characterized via the Mooney–Rivlin strain energy model and the fiber kinematics is computed via the first and second gradient of deformations. By applying the variational principle on the strain energy of FRC, the Euler–Lagrange equilibrium equations are derived and then solved numerically. The theoretical results highlight the matrix and meshwork deformations of FRC subjected to bilateral bending and stretching simultaneously, and it is found that the interaction between bilateral extension and bending manipulates the matrix and network deformation. It is theoretically observed that the transverse Lagrange strain peaks near the bilateral boundary while the longitudinal strain is intensified inside the FRC domain. The continuum model further demonstrates the bidirectional mesh network deformations in the case of plain woven, from which the extension and flexure kinematics of fiber units are illustrated to examine the effects of fiber unit deformations on the overall deformations of the fiber network. To reduce the observed matrix-network dislocation in the case of plain network reinforcement, the pantographic network reinforcement is investigated, suggesting that the bilateral stretch results in the reduced intersection angle at the mesh joints in the FRC domain. For validation of the continuum model, the obtained results are cross-examined with the existing experimental results depicting the failure mode of conventional fiber-reinforced composites to demonstrate the practical utility of the proposed model. Full article

Cell membranes contain a variety of biomolecules, especially various kinds of lipids and proteins, which constantly change with fluidity and environmental stimuli. Though Helfrich curvature elastic energy has successfully explained many phenomena for single-component membranes, a new theoretical framework for multicomponent membranes is still a challenge. In this work, we propose a generalized Helfrich free-energy functional describe equilibrium shapes and phase behaviors related to membrane heterogeneity via curvature-component coupling within a unified framework. For multicomponent membranes, a new but important Laplace–Beltrami operator is derived from the variational calculation on the integral of Gaussian curvature and applied to explain the spontaneous nanotube formation of an asymmetric glycolipid vesicle. Therefore, our general mathematical framework shows predictive capabilities beyond the existing multicomponent membrane models. A set of new curvature-component coupling Euler-Lagrange equations has been derived for global vesicle shapes associated with the composition redistribution of multicomponent membranes for the first time and specified into several typical geometric shapes. The equilibrium radii of isotonic vesicles for both spherical and cylindrical geometries are calculated. The analytical solution for isotonic vesicles reveals that membrane stability requires distinct bending rigidities among components ($k_{\mathrm{A}}\ne k_{\mathrm{B}}$, ${\overline{k}}_{\mathrm{A}}\ne {\overline{k}}_{\mathrm{B}}$) whose bending rigidities are linearly related, which is consistent with experimental observations of coexisting lipid domains. Furthermore, we elucidate the biophysical implications of the derived shape equations, linking them to experimentally observed membrane remodeling processes. Our new free-energy framework provides a baseline for more detailed microscopic membrane models. Full article

The presented study demonstrates that UAVs can be flown with a morphing wing to develop essential aerodynamic efficiency without a tail structure, which decides the operational cost and flight safety. The mechanical control for morphing is discussed, where the system design, simulation, and experimental realization of ±15° SDOF sweep motion for a 7 kg eVTOL wing are detailed. The methodology, developed through a mathematical modeling of the mechanism’s kinematics and dynamics, is explained using Denavit–Hartenberg (D-H) convention, Lagrangian mechanics, and Euler–Lagrangian equations. The simulation and MBD analyses were performed in MATLAB R2021 and by Altair Motion Solve, respectively. The experiment was conducted on a dedicated test rig with two wing variants fitted with IMUs and an autopilot. The results from various methods were analyzed and experimentally compared to provide an accurate insight into the system’s design, modeling, and performance of the sweep morphing wing. The theoretical calculations by the mathematical model were compared with the test results. The sweep requirement is essential for eVTOL to have long endurance and multi-mission capabilities. Therefore, the developed sweep morphing mechanism is very useful, meeting such a demand. However, the results for three-dimensional morphing, operating sweep, pitch, and roll together are also presented, for the sake of completeness. Full article

The vibration control in structural design has long been a critical area of study, particularly in mitigating undesirable resonant vibrations using dynamic vibration absorbers (DVAs). Traditional approaches to tuning DVAs have relied on complex mathematical models based on Newtonian or Euler–Lagrange equations, often leading to intricate systems requiring simplification of the analysis of multi-degree-of-freedom structures. This paper introduces a novel modeling approach for analyzing DVAs based on the concept of global admittance, which stems from mechanical admittance and network simplifications. This model streamlines the representation of structures with DVAs as one-degree-of-freedom systems coupled with a global admittance function, which emulates additional damping coupled to the primary structure. In this work, global admittance functions are determined by the independent analysis of the mechanical networks of the DVA, restructuring the process of obtaining the system’s transfer function. The model was validated using different classical DVA configurations, demonstrating total accuracy in its applicability across designs concerning conventional modeling. Our most remarkable finding was that the dimensionless function, ${\gamma }_g\left(\mathsf{\Omega }\right)$, resulting from the global admittance, partially decouples the dynamics of the DVAs from the primary structure, facilitating the implementation of passive vibration control strategies in more realistic structural models. Additionally, this work establishes a significant advancement in vibration control analysis, providing a flexible tool for control strategies in real-world structural systems. Full article

This paper extends the fractional calculus of variations to include generalized fractional derivatives with dependence on a given kernel, encompassing a wide range of fractional operators. We focus on variational problems involving the composition of functionals, deriving the Euler–Lagrange equation for this generalized case and providing optimality conditions for extremal curves. We explore problems with integral and holonomic constraints and consider higher-order derivatives, where the fractional orders are free. Full article

The Euler–Lagrange and Newton–Euler methods are typically used to derive equations of motion for serial-link manipulators. We previously proposed a partial Lagrangian method, which is similar to the Lagrangian method, for handling the equations of motion analytically. Moreover, the proposed method can efficiently handle multi-link analyses, similar to the Newton–Euler method. The partial Lagrangian method organizes the Lagrangian, which is obtained from the link structure, and torque, which is obtained by differential operations, into a table that can be easily handled by both manual calculations and computer analysis. Furthermore, by representing it using a computational graph, it is possible to perform dynamic analysis while maintaining the structure of a system. By observing the intermediate nodes of this computational graph, it is possible to observe how the torque generated at a particular link affects the joint. Organizing the structure with graphs allows us to consider complex systems as a collection of subgraphs, making this method highly compatible with our proposed partial Lagrangian approach. This study shows that the partial torque tensor can be used as an analog of the partial torque table for serial-link systems by interpreting the meaning of the table, i.e., the partial torque, as the interaction between the links in order to simplify the treatment of branching link systems. Due to the use of the partial torque tensor, the dimensions of the tensor correspond one-to-one with the number of branches, allowing the description of any branching system. Furthermore, by using the proposed building block representation, even complex branching systems can be easily designed and analyzed. Full article