Search Results (465)

Aiming at the trajectory tracking control problem caused by the coupling of strong nonlinearity, parameter uncertainty and unknown disturbances in rigid robotic arms, this paper proposes an adaptive robust model predictive control (APRMPC) scheme. This study aims to fill the gap in the existing literature by proposing a dedicated control framework capable of simultaneously and effectively handling parameter uncertainty, unmodeled dynamics, and external disturbances, while ensuring constraint satisfaction. Firstly, a dynamic model of a three-degree-of-freedom robotic arm was established based on the Lagrange equation; secondly, this paper designs a deep integration mechanism of adaptive law and robust predictive control: by designing a parameter adaptive algorithm to estimate the system uncertainty online and feedforward compensate it to the predictive model, the impact of model mismatch is significantly reduced; meanwhile, for the estimated residuals and unknown disturbances, feedback gain was introduced and the control input was designed based on the robust invariant set theory, achieving unified parameter identification, disturbance suppression and rolling optimization within a single framework. This paper strictly proves the feasibility and stability of the control scheme. Finally, the simulation experiments based on MATLAB show that, compared with the traditional MPC and PID methods, the APRMPC algorithm can achieve higher accuracy and stronger robustness in trajectory tracking under various working conditions, effectively resolving the inherent contradiction between the weak robustness of the traditional MPC and the large buffering of sliding mode control, and verifying the value of the proposed scheme in filling the gap in related literature. Full article

The design, control, simulation and animation of robotic systems heavily depend on dynamic modeling. A variety of studies have explored different dynamic modeling methodologies applied to diverse robotic mechanisms. Artificial neural networks (ANNs) have proven their value in engineering design in recent years, enhancing the understanding of complex mechanisms as well as shortening experimental periods and decreasing related expenses. This study investigates the application of various neural network algorithms for the analysis of a custom-designed three-link planar revolute–prismatic–revolute (RPR) robotic arm mechanism. Initially, the Euler–Lagrange equations of motion for the RPR mechanism are derived. Joint accelerations are then computed under different mass configurations of the robotic links, resulting in a dataset comprising 204 joint acceleration samples. Six distinct neural network models are subsequently employed to perform regression analysis on the collected data. The primary objective of this study is to analyze the relationship between joint accelerations and varying link masses under constant joint torques and forces, while its secondary aim is to present a representative application of neural networks as regression learners for the dynamic modeling of robotic mechanisms. The approach outlined in this study allows users to select appropriate neural network algorithms for use in specific applications, considering the wide range of available algorithms. Link mass variations and their effects on joint accelerations are investigated, establishing a basis for the modeling of robotic dynamics using regression-based neural networks. The results indicate that the optimizable neural network algorithm produces the best regression accuracy results, although the other models maintain similar performance levels. Full article

The Search and Rescue at Sea Manual defines several uncertainties related to the initial position and the time elapsed between an accident and the onset of SAR operations. The present article seeks an approach to address this problem through the assimilation of drifting buoy data and their use in correcting the system parameters via an ill-posed inverse problem. The results demonstrate that, in the search for objects at sea, the uncertainty of their initial position must be explicitly considered. Quantitatively, the proposed methodology reduced the uncertainty of the initial search area by approximately 55–60% compared with the traditional approach that assumes a single deterministic initial point. This outcome underscores the potential of data assimilation techniques to enhance the probabilistic accuracy of maritime search and rescue planning. Full article

In this study, a nonlinear dynamic model is developed to examine the stability and vibration behavior of a self-balancing electric Segway operating over irregular stochastic terrains. The Segway is treated as a three-degrees-of-freedom cart–inverted pendulum system, incorporating elastic and damping effects at the wheel–ground interface. Road irregularities are generated in accordance with international standard using high-order filtered noise, allowing for representation of surface classes from smooth to highly degraded. The governing equations, formulated via Lagrange’s method, are transformed into a Lorenz-like state-space form for nonlinear analysis. Numerical simulations employ the fourth-order Runge–Kutta scheme to compute translational and angular responses under varying speeds and terrain conditions. Frequency-domain analysis using Fast Fourier Transform (FFT) identifies resonant excitation bands linked to road spectral content, while Kernel Density Estimation (KDE) maps the probability distribution of displacement states to distinguish stable from variable regimes. The Lyapunov stability assessment and bifurcation analysis reveal critical velocity thresholds and parameter regions marking transitions from stable operation to chaotic motion. The study quantifies the influence of the gravity–damping ratio, mass–damping coupling, control torque ratio, and vertical excitation on dynamic stability. The results provide a methodology for designing stability control systems that ensure safe and comfortable Segway operation across diverse terrains. Full article

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

In this paper, we present a transient-format time-continuous Galerkin finite element method for fully intrinsic geometrically exact beam equations that are energy-consistent. Within the grid of space and time, we derive governing equations for elements using the Galerkin method and the time finite element method, implement variable interpolation via Legendre functions, and establish an assembly process for space–time finite element equations. The key achievement is the realization of the free order variation of the program, which provides a basis for future research on adaptive algorithms. In particular, the variable order method reduces the quality requirements for the mesh. In regions with a higher degree of nonlinearity, it is easier to increase the variable order, and the result is smoother. Meanwhile, increasing the interpolation order effectively enhances computational accuracy. Introducing kinematical equations of rotation with Lagrange operators completely imposes the conservative loads on fully intrinsic equations. This means that loads in the inertial coordinate system, such as gravity, can also be iterated synchronously in the deformed coordinate system. Through a set of illustrative examples, our algorithm demonstrates effectiveness in addressing conservative loads, elastic coupling deformation, and dynamic response, demonstrating the ability to analyze elastically coupled dynamic problems pertaining to helicopter rotors. 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

The inchworm piezoelectric motor, with the advantages of long stroke and high resolution, is ideally suited for precise positioning in wafer-level electron beam inspection systems. However, the large number of piezoelectric actuators and the complex excitation signal sequences significantly increase the complexity of system assembly and temporal control. A flexure-based actuation stator structure, along with simplified excitation signal sequences of a high-precision inchworm piezoelectric motor, is proposed. The alternating actuation of upper/lower clamping mechanisms and the driving mechanism fundamentally mitigates backstep effects while generating stepping linear displacement. The inchworm piezoelectric motor achieves precision linear motion operation using only two piezoelectric actuators. The actuation stator is analyzed via the compliance matrix method to derive its output compliance, input stiffness, and displacement amplification ratio. Furthermore, a kinematic model and natural frequency expression incorporating the pseudo-rigid-body method and Lagrange’s equations are established. The actuation stator and inchworm piezoelectric motor are analyzed through both simulations and experiments. The results show that the maximum step displacement of the motor is 16.3 μm, and the maximum speed is 9.78 mm/s, at a 600 Hz operation frequency with a combined alternating piezoelectric voltage of 135 V and 65 V. These findings validate the designed piezoelectric motor’s superior motion resolution, operational stability, and acceptable load capacity. 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

Atmospheric climate science lacks the capacity to integrate thermodynamics with the gravitational potential of air in a classical quantum theory. To what extent can we identify Carnot’s ideal heat engine cycle in reversible isothermal and isentropic phases between dual temperatures partitioning heat flow with coupled work processes in the atmosphere? Using statistical action mechanics to describe Carnot’s cycle, the maximum rate of work possible can be integrated for the working gases as equal to variations in the absolute Gibbs energy, estimated as sustaining field quanta consistent with Carnot’s definition of heat as caloric. His treatise of 1824 even gave equations expressing work potential as a function of differences in temperature and the logarithm of the change in density and volume. Second, Carnot’s mechanical principle of cooling caused by gas dilation or warming by compression can be applied to tropospheric heat–work cycles in anticyclones and cyclones. Third, the virial theorem of Lagrange and Clausius based on least action predicts a more accurate temperature gradient with altitude near 6.5–6.9 °C per km, requiring that the Gibbs rotational quantum energies of gas molecules exchange reversibly with gravitational potential. This predicts a diminished role for the radiative transfer of energy from the atmosphere to the surface, in contrast to the Trenberth global radiative budget of ≈330 watts per square metre as downwelling radiation. The spectral absorptivity of greenhouse gas for surface radiation into the troposphere enables thermal recycling, sustaining air masses in Lagrangian action. This obviates the current paradigm of cooling with altitude by adiabatic expansion. The virial-action theorem must also control non-reversible heat–work Carnot cycles, with turbulent friction raising the surface temperature. Dissipative surface warming raises the surface pressure by heating, sustaining the weight of the atmosphere to varying altitudes according to latitude and seasonal angles of insolation. New predictions for experimental testing are now emerging from this virial-action hypothesis for climate, linking vortical energy potential with convective and turbulent exchanges of work and heat, proposed as the efficient cause setting the thermal temperature of surface materials. Full article

This paper studies the existence, regularity, and properties of normalized ground state solutions for the mixed fractional Schrödinger equations. For subcritical cases, we establish the boundedness and Sobolev regularity of solutions, derive Pohozaev identities, and prove the existence of radial, decreasing ground states, while showing nonexistence in the $L^2$-critical case. For $L^2$-supercritical exponents, we identify parameter regimes where ground states exist, characterized by a negative Lagrange multiplier. The analysis combines variational methods, scaling techniques, and the careful study of fibering maps to address challenges posed by competing nonlinearities and nonlocal interactions. Full article