Inverse Dynamics Problem Research Articles

Families of Planar Orbits in Polar Coordinates Compatible with Potentials

In light of the planar inverse problem of Newtonian Dynamics, we study the monoparametric family of regular orbits f(r,θ)=c in polar coordinates (where c is the parameter varying along the family of orbits), which are generated by planar potentials V=V(r,θ). The corresponding family of orbits can be uniquely represented by the “slope function” γ=fθfr. By using the basic partial differential equation of the planar inverse problem, which combines families of orbits and potentials, we apply a new methodology in order to find specific potentials, e.g., V=A(r)+B(θ) or V=H(γ) and one-dimensional potentials, e.g., V=A(r) or V=G(θ). In order to determine such potentials, differential conditions on the family of orbits f(r,θ) = c are imposed. If these conditions are fulfilled, then we can find a potential of the above form analytically. For the given families of curves, such as ellipses, parabolas, Bernoulli’s lemniscates, etc., we find potentials that produce them. We present suitable examples for all cases and refer to the case of straight lines.

Inverse chance-constrained dynamic data envelopment analysis under natural and managerial disposability: Concerning renewable energy efficiency and potentials

The performance of dynamic systems with specific data has been examined in some data envelopment analysis (DEA) studies. However, in many real-life situations, dealing with dynamic systems becomes challenging because of random factors. So, in this paper, a chance-constrained dynamic DEA approach under managerial and natural disposability is first proposed to analyze the dynamic renewable energy efficacy of processes in the presence of random performance measures. Then, the potentials of some performance metrics are addressed for changes of others using the introduced inverse chance-constrained dynamic DEA models. The chance-constrained dynamic DEA techniques and their inverse dynamic stochastic problems are converted into linear problems. The introduced approaches are applied to probe the renewable energy efficiency and potentials of some OECD countries in a span of time. The results show stochastic dynamic DEA and inverse stochastic dynamic DEA provided are practical tools for making the best choices, when there is uncertainty.

Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems

This study introduces an innovative neural network framework named spectral integrated neural networks (SINNs) to address both forward and inverse dynamic problems in three-dimensional space. In the SINNs, the spectral integration technique is utilized for temporal discretization, followed by the application of a fully connected neural network to solve the resulting partial differential equations in the spatial domain. Furthermore, the polynomial basis functions are employed to expand the unknown function, with the goal of improving the performance of SINNs in tackling inverse problems. The performance of the developed framework is evaluated through several dynamic benchmark examples encompassing linear and nonlinear heat conduction problems, linear and nonlinear wave propagation problems, inverse problem of heat conduction, and long-time heat conduction problem. The numerical results demonstrate that the SINNs can effectively and accurately solve forward and inverse problems involving heat conduction and wave propagation. Additionally, the SINNs provide precise and stable solutions for dynamic problems with extended time durations. Compared to commonly used physics-informed neural networks, the SINNs exhibit superior performance with enhanced convergence speed, computational accuracy, and efficiency.

The ensemble kalman filter for dynamic inverse problems

The ensemble kalman filter for dynamic inverse problems

Inverse problem of dynamics, Galiullin and Szebehely methods and curl force trajectories

At first we study Galiullin’s construction of Bertrand problem and compare it with Szebehely’s method, latter is based on a first order partial differential equation for the unknown potential that produces a prescribed monoparametric family of planar trajectories. In the second part of the paper we study the inverse problem of the trajectories such that the corresponding force is a nonconservative position dependent one, satisfying the non-vanishing curl condition and not the gradient of a potential function. Recently this force has been introduced and popularized by Berry and Shukla (J. Phys. A 45 (2012) 305 201). We connect the inverse problem dynamics of these curl force trajectories with the generalized potentials obtained by Sarlet-Mestdag-Prince (Rep. Math. Phys. 72(2013) 65-84) from the inverse problem of ϕ(x, y) = xy m for m ≠ 0, m ≠ –1. Finally we show that the analog of these curly trajectories in momentum space can be manifested as kinetic energies of the pair of Calogero-Leyvraz Hamiltonians (J. Nonlinear Math. Phys. 26 (2019) 147-154) describing the motion of a particle in a magnetic field with friction.

Variable coefficient-informed neural network for PDE inverse problem in fluid dynamics

Variable-coefficient equations are crucial in the field of fluid dynamics as they accurately capture the spatial and temporal variations of fluid properties. In many cases, there exist some constraints among multiple coefficients and embedding these constraints into neural networks poses a challenge. In this paper, we design a variable coefficient-informed neural network (VCINN) to address the inverse problem of variable-coefficients partial differential equation in fluid dynamics. The VCINN framework integrates the physics-informed neural network (PINN) with the constraints among multiple coefficients, encoding both constraints and physics information into the neural networks. Compared to classical PINN, VCINN enjoys such several advantages as parallelization capacity, embedding constraint information and efficient hyperparameter adjustment. Through series of examples, the capability of the approach to recover coefficients from observations has been demonstrated. Numerical results indicate that the present method achieves higher accuracy and lower training error compared to classical PINN.

Integrated control‐structure co‐design for flexible manipulators

AbstractThis work aims to develop a methodology for the integrated control‐structure design of flexible multibody systems. Such an issue is linked to a generalization of the inverse dynamics problem, where both the feedforward control and mechanical parameters are designed to perform a desired trajectory optimally. The resulting constrained motion is aimed at improving the dynamic performance of the system, which is underactuated, with less power demand but possible elastic deformation. Stable inversion methods are required to deal with the internal dynamics related to underactuation. The proposed methodology is based on the optimal control theory. For illustrative purposes, we considered a fully actuated linear time‐invariant (LTI) system, establishing the integrated design results in a least‐squares homogeneous system and in an optimization problem. An additional formulation is proposed for the integrated design of the flexible multibody systems, where both a multi‐criterion optimization and a scalarized problem of a corresponding nonlinear programming problem are formulated to deal with the unstable zero‐dynamics related to non‐minimum phase systems and nonlinearity. The numerical simulation of an underactuated two‐masses‐spring‐damper system and a flexible robotic manipulator in planar motions are used for validation. Finally, the beneficial aspects of the integrated design are analyzed.

Nonlinearity in the inverse problems of orbital dynamics using the example of potentially hazardous asteroids and outer satellites of Jupiter

Nonlinearity in the inverse problems of orbital dynamics using the example of potentially hazardous asteroids and outer satellites of Jupiter

Formation of Flight Trajectories of an Unmanned Glider Based on the Methodology of Inverse Dynamics Problems

Formation of Flight Trajectories of an Unmanned Glider Based on the Methodology of Inverse Dynamics Problems

Solvability of a One-Dimensional Dynamic Direct and Inverse Problem for a System of Hopf Type Equations

Solvability of a One-Dimensional Dynamic Direct and Inverse Problem for a System of Hopf Type Equations

A data-driven force-thermal coupling load identification method considering multi-source uncertainties of structural characteristics and measuring noises

The problem of load identification denotes identifying loads based on the measurement of structural responses, which is the inverse problem in structural dynamics. With advancements in aeronautical technology, the working environment of aircraft becomes even more complex. To accurately monitor and forecast the load environment where the structure works can provide guidance for the structural design of aircraft and predict potential structural damage. However, with the development of hypersonic aircraft, the influence of thermal field also cannot be ignored besides external forces that are applied to the structure, thus, there is an urgent need for effective methods to identify both force and thermal loads. In this paper, firstly, A data-driven load identification method and various load identification strategies for the identification of force-thermal load based on Artificial Neural Networks are proposed, and in different loading cases the relative error of identification is less than 5 %, with the training process converging efficiently within a short time; additionally, multi-source uncertainties, including Gaussian white noises and structural uncertainties, are simulated, and their influence on load identification is also evaluated; moreover, sensor placement optimization based on particle swarm optimization is carried out to enhance accuracy of load identification, and the number of sensors and the corresponding optimal placement of sensors are determined, it can be proved via numerical examples that the optimized sensor placement can reduce the error of load identification by more than 90 % of that of a random sensor placement.

Математическая модель тренажера «конь – наездник» и ее программная реализация в системе компьютерной математики Wolfram Mathematica

Riding simulation machines allow training horse-back riders regardless of environmental and weather conditions, horse fatigue, stable remoteness and other factors. Riders can practice complex movements without fear of potential injury related to sometimes unpredictable behavior of the horse. This type of training machines requires special software for selecting the relevant action mode for the person during the rider’s practice. Thus, the selected behavior of the horse can be simulated and the injuries of horse-back rider can be avoided. For this purpose, the model of horse-back rider, taking into account its training level and the horse motion, is required. The model of “person – simulation machine” combination is proposed in the article. The proposed model allows developing operating modes taking into account the sportsman-training machine interaction. The mechanical model is an exosuit with a mobile pole, consisting of four links, the rider’s feet, shins, hips, and the corps are attached to. The pole corresponds to the mass center of the horse, the rider interacts with by standing on the stirrups. The pole motion is implemented by a telescopic link attached to immobile base. The relative rotations of the links are implemented by cylindrical hinges with negligible friction. The proposed model has been implemented as software in the “Wolfram Mathematica 11.3” environment. It has been designed for training machine dynamics simulation of the “horse ­– rider” system. The software includes several modules: 1) the module for specifying the structure of training machine mathematical model, generalized coordinates, and for auto-compilation of the corresponding system of differential equations; 2) the module for specifying the programmed model motion and calculation of the required control torques in the hinges; 3) the module for the Cauchy problem numerical solution; 4) the module for the animated visualizing of the model motion, and for exporting the obtained graphic results and numerical calculations. The developed software allows conducting dynamics analysis of mathematical model for the considered system based on the solution of both direct and inverse dynamics problems. Also it can be suggested for designing training machines with programmed operation mode. It has been shown that application of this software accelerates the development of training machines.

Semi-Separable Potentials as Solutions to the 3D Inverse Problem of Newtonian Dynamics

We study the motion of a test particle in a conservative force-field. Our aim is to find three-dimensional potentials with symmetrical properties, i.e., V(x,y,z)=P(x,y)+Q(z), or, V(x,y,z)=P(x2+y2)+Q(z) and V(x,y,z)=P(x,y)Q(z), where P and Q are arbitrary C2-functions, which are characterized as semi-separable and they produce a pre-assigned two-parametric family of orbits f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const) in 3D space. There exist two linear PDEs which are the basic equations of the Inverse Problem of Newtonian Dynamics and are satisfied by these potentials. Pertinent examples are presented for all the cases. Two-dimensional potentials are also included into our study. Families of straight lines is a special category of curves in 3D space and are examined separately.

Real and Complex Potentials as Solutions to Planar Inverse Problem of Newtonian Dynamics

We study the motion of a test particle in a conservative force field. In the framework of the 2D inverse problem of Newtonian dynamics, we find 2D potentials that produce a preassigned monoparametric family of regular orbits f(x,y)=c on the xy-plane (where c is the parameter of the family of orbits). This family of orbits can be represented by the “slope function” γ=fyfx uniquely. A new methodology is applied to the basic equation of the planar inverse problem in order to find potentials of a special form, i.e., V=F(x+y)+G(x−y), V=F(x+iy)+G(x−iy) and V=P(x)+Q(y), and polynomial ones. According to this methodology, we impose differential conditions on the family of orbits f(x,y) = c. If they are satisfied, such a potential exists and it is found analytically. For known families of curves, e.g., circles, parabolas, hyperbolas, etc., we find potentials that are compatible with them. We offer pertinent examples that cover all the cases. The case of families of straight lines is referred to.

EN Fluid dynamic design of centrifugal compressor diffusers

Diffusers are used to convert the kinetic energy of the gas flow into potential energy, i.e. to reduce the velocity and to increase the pressure. The most common types of diffusers are vaneless, vaned, and channel ones. Each type of diffuser has its design features and its performances. Improving the operational cha­racteristics of vaneless diffusers is possible by using stepped diffusers with flow injection. The developed mathematical model of gas flow in a vaneless diffuser with flow injection provides design of the stage of centrifugal compressor with the wider range of stable operation. Traditional methods of designing vaned and channel diffusers of turbomachines are focused on the use of simple geometric lines and surfaces, such as a straight line, arc of a circle, plane, cylindrical surface, etc. The design is performed according to recommendations based on experimental data. The other way of designing the vaned and channel diffusers of turbomachines is related to solving the inverse problem of fluid dynamics, when the shape of the surfaces is determined by the given distribution of velocities along the surfaces of the channel. This paper describes the design principles for the centrifugal compressor diffusers based on physical and mathema­tical models of the flow of swirling viscous compressible fluid. According to the presented method, the designing diffusers are based on the preseparation condition of the boundary layer along one of the surfa­ces. Such a design ensures a reduction in separation zones in the channels of the diffusers and, accor­dingly, a reduction in total pressure losses and an expansion of the range of stable operation. A new method of designing the vaned and channel diffusers provides an improvement in the gasdynamic characteristics of diffusers compared to traditional geometry diffusers

Dynamical programming for off-the-grid dynamic inverse problems

In this work we consider off-the-grid algorithms for the reconstruction of sparse measures from time-varying data. In particular, the reconstruction is a finite collection of Dirac measures whose locations and masses vary continuously in time. Recent work showed that this decomposition was possible by minimising a convex variational model which combined a quadratic data fidelity with dynamical Optimal Transport. We generalise this framework and propose new numerical methods which leverage efficient classical algorithms for computing shortest paths on directed acyclic graphs. Our theoretical analysis confirms that these methods converge to globally optimal reconstructions. Numerically, we show new examples for unbalanced Optimal Transport penalties, and for balanced examples we are 100 times faster in comparison to the previously known method.

ДЕЯКІ КРАЙОВІ ЗАДАЧІ ДРОБОВО-ДИФЕРЕНЦІЙНОЇ ФІЛЬТРАЦІЙНОЇ ДИНАМІКИ ЩОДО БІПАРАБОЛІЧНОЇ МАТЕМАТИЧНОЇ МОДЕЛІ

Closed-form solutions are obtained to some one-dimensional boundary-value problems for modeling anomalous filtration dynamics in a layered geoporous medium, posed within the framework of the fractional-differential generalization of the biparabolic evolutionary partial differential equation of the fourth order. In particular, the formulation and solution of the direct and inverse model boundary-value problems of geofiltration dynamics based on the mathematical model with conjugation conditions are presented, and the conditions of the existence of regular solutions to these problems are defined. Keywords: mathematical modeling, fractional-differential dynamics of geofiltration processes, nonclassical models, biparabolic evolutionary equation, the fractional-differential analog of the biparabolic evolutionary equation, nonstationary boundary-value problems on a finite interval, direct and inverse problems, conjugation conditions, closed-form solutions.

Analysis of Possible Solutions of Some Inverse Dynamical Problem with Regard for Constraint Stabilization

Analysis of Possible Solutions of Some Inverse Dynamical Problem with Regard for Constraint Stabilization

An improved algorithm for solving inverse problems of the dynamics of robotic devices at SCAS KIDYM

The paper presents the results of studies of an analytical algorithm for constructing a resolving system of linear equations to determine the driving forces and moments in the links of robotic devices of arbitrary structure during movement. This algorithm is implemented as an analytical algorithm in a special computer algebra system (SCAS) KiDyM, on the basis of which numerical calculations of the specified forces and moments are generated in the process of performing work processes, positioning mechanisms in the required positions, parking movements, etc. The KiDyM software package is used to solve mechanics problems of a wide class of discrete mechanical systems of arbitrary structure and type of motion for engineering and scientific calculations. A theoretical justification for the developed new approach, analytical and numerical proof of its effectiveness from the point of view of the process of automatically obtaining resolving equations, computational efficiency and accuracy of results are presented. The proposed algorithm is the result of the development of an algorithm that was previously implemented in SCAS KiDyM and was similar in purpose. Unlike the previously implemented algorithm, which generates a resolving system of equations after constructing the dynamic equations, the new algorithm is built into the process of generating dynamic equations - it sorts the model elements - inertial, dissipative, elastic and force elements of known forces and moments form the right side, and force elements with unknown forces and moments, they form a matrix of the left side of equations that solve the inverse problem of dynamics. The article gives a comparative analytical conclusion of one and another algorithm and shows the coincidence of the resulting equations. Also, using examples of calculations of the inverse problem of the dynamics of a portal crane and two 6-degree manipulators, the advantages of the new approach are shown.

INVERSE AND FORWARD KINEMATICS AND DYNAMICS OF A TWO LINK ROBOT ARM

This paper deals with the kinematic and dynamic analysis of a two-element robot model. We have solved the problem using inverse kinematics. The result is the plotted trajectory of the robot's end point along defined points of the robot's workspace. The angular quantities computed are the rotation angle, angular velocity and angular acceleration in each kinematic pair. Furthermore, an inverse dynamics problem is solved. The moments in the kinematic pairs are computed. Subsequently, using the direct dynamics problem, the correctness of the obtained solution is confirmed. The programs designed for the simulation of dynamical systems - Matlab/Simulink and SimMechanics - were used in the solution. The results are presented in the form of graphs and tables.