Inverse Dynamics Problem Research Articles

Matrix solution for the inverse problem of classical dynamics of a particle

We solve the inverse problem corresponding to the fundamental problem of the classical dynamics of a material particle through a matrix treatment: assuming knowing the mass and the position (the trajectory, in relation to an inertial reference) of a particle at all times, we impose that this corresponds to the eigenvector of a “position matrix". Subsequent development leads to a “force matrix", which has the resultant force on the particle as its eigenvector. We identified some limitations of this matrix treatment.

3D homogeneous potentials generating two-parametric families of orbits on the outside of a material concentration

We study three-dimensional homogeneous potentials V = V(x, y, z) of degree m which are created outside a finite concentration of matter and they produce a preassigned two-parametric family of spatial regular orbits given in the solved form f(x, y, z) = $$c_{1}$$ , g(x, y, z) = $$c_{2}$$ ( $$c_{1}$$ , $$\; c_{2}$$ = {\rm const}). These potentials have to satisfy three linear PDEs; two of them come from the Inverse Problem of Newtonian Dynamics and the last one is the well-known ”Laplace’s equation”. Our aim is to find common solutions for these three PDEs. Besides that we consider that the functions f and g are also homogeneous in the variables $$x,\; y,\;z$$ of any degree and can be represented uniquely by the ”slope functions” $$\alpha (x,y,z)$$ and $$\beta (x,y,z)$$ which are homogeneous of zero degree. Then, we impose three differential conditions on the orbital functions ( $$\alpha , \; \beta $$ ). If they are satisfied for a specific value of m, then we can find the potential by quadratures. The values obtained for m so far are consistent with familiar gravitational and electrostatic and quadratic potentials. Finally, pertinent examples are given and cover all the cases.

Maximumly weighted iteration for solving inverse problems in dynamics

Many problems in dynamics can be formulated as inverse problems that require the determination of the unknown input from the known output. Limited by the measurement noise, the computational cost, the error in numerical computation, etc., solving ill-conditioned inverse problems in dynamics is a big challenge. In this paper, we propose Maximumly Weighted Iteration (MWI) approach to solve ill-conditioned inverse problems in dynamics generally. The ill-condition of the system coefficient matrix is controlled by iterative weighted decomposition and a weighted term, also avoiding matrix inversion of solving and reconstruction process. As an iterative method, MWI has been theoretically proven to have consistency, absolute convergence, and conditional unbiasedness. The numerical results demonstrate that MWI is superior to Truncated Singular Value Decomposition and Tikhonov regularization in terms of accuracy and anti-noise property. Two application cases are used to demonstrate the potential of MWI in engineering practice. The first is elastic vibrations identification combined with randomized resonant metamaterials, and the second is fault diagnosis of mechanical transmission systems.

Creation of an electromechanical model of an exoskeleton link in the form of Lagrange-Maxwell equations for agricultural mechanization

The electromechanical model of exoskeleton link has been developed for agricultural mechanization. First, a differential equation of link motion was compiled without taking into account electric drives in the form of the Lagrange equation of the second kind. For it, the inverse and direct problems of dynamics are solved. Then, this equation takes into account the influence of the rotating rotor and the gear ratio of the gearbox on the dynamics of the link. For this model, the direct problem of dynamics is solved. A significant influence on the results of the numerical solution for taking into account the rotating rotor of the electric drive has been established. Then the system of differential equations, describing dynamics of the controlled motion of the model, in the form of Lagrange-Maxwell equations has been composed. The local system of coordinates has been applied in the model, since the electric drive changes the angles between the links. Direct and inverse dynamics problems have been solved. The comparative analysis of the models with electric drive and without it has been made. It has been established that taking into account the electrical drive system allows achieving good results in modeling accuracy.

A computational framework for edge-preserving regularization in dynamic inverse problems

A computational framework for edge-preserving regularization in dynamic inverse problems

Identification and dynamics prediction of a plane vortex structure based on a mathematical model of a point vortices system

The aim of the article is developing and analyse an algorithmic method for solution finding of one inverse problem of 2d vortex fluid dynamics. It is identification and prediction of the flow structure evolution of the based on the data on fluid velocity vectors in a set of reference points. Theoretical analysis of convergence and adequacy of the method is difficult due to the ill-posedness typical of inverse problems, these issues studied experimentally. Methods. The proposed method uses a mathematical model of a point vortex dynamics system for identification and prediction flow structures. The parameters of the model system are found by minimising the functional that evaluates the closeness of the original and model vectors fields at the reference points. The prediction of the vortex structure dynamics is based on the solution of the Cauchy problem for a system of ordinary differential equations with the parameters found in the first stage. Results. As a result of the calculations, we found it out: the algorithm converges to the desired minimum from a wide range of initial approximations; the algorithm converges in all cases when the identified structure consists of sufficiently distant vortices; the forecast of the development of the current gives good results with a steady flow; if the above conditions are violated, the part of successful calculations decreases, false identification and an erroneous forecast may occur; with the convergence of the method, the coordinates and circulation of the eddies of the model system are close to the characteristics of the eddies of the test configurations; the structures of the streamlines of the flows are topologically equivalent; convergence depends more on location than on the number of vectors used for identification. Conclusion. An algorithm for solving the problem of identifying and the evolution forecast of a 2d vortex flow structure is proposed when the fluid velocity vectors in a finite set of reference points are known. The method showed its high efficiency when using from 40 to 200 reference points. The results of the study make it possible to recommend the proposed algorithm for identifying flat vortex structures, which consist of vortices separated from each other.

Описание программного комплекса для моделирования робота-манипулятора

The article proposes the development of a software module for modeling the kinematics and dynamics of a manipulator with five degrees of freedom. To solve the forward kinematics problem of the manipulator, the Denavit–Hartenberg method was used. To solve the inverse kinematics and dynamics problem of the manipulator, analytical methods (the Levenberg-Marquardt method, the Newton–Euler method) and a soft computing method (adaptive neurofuzzy inference system) were used. The software module for modeling the kinematics and dynamics of the manipulator was developed using the software package of the SolidWorks computer-aided design system and the MatLab program. The developed software module is able to simulate the kinematics and dynamics of the manipulator based on the described methods, visualize the simulation results, generate a trajectory for the target position and orientation of the end-effector of the manipulator, simulate the movement of the manipulator along a given trajectory.

Projecting basis functions with tensor networks for Gaussian process regression

This paper presents a method for approximate Gaussian process (GP) regression with tensor networks (TNs). A parametric approximation of a GP uses a linear combination of basis functions, where the accuracy of the approximation depends on the total number of basis functions M. We develop an approach that allows us to use an exponential amount of basis functions without the corresponding exponential computational complexity. The key idea to enable this is using low-rank TNs. We first find a suitable low-dimensional subspace from the data, described by a low-rank TN. In this low-dimensional subspace, we then infer the weights of our model by solving a Bayesian inference problem. Finally, we project the resulting weights back to the original space to make GP predictions. The benefit of our approach comes from the projection to a smaller subspace: It modifies the shape of the basis functions in a way that it sees fit based on the given data, and it allows for efficient computations in the smaller subspace. In an experiment with an 18-dimensional benchmark data set, we show the applicability of our method to an inverse dynamics problem.

Development of control algorithms for magnetoelectric generator with axial magnetic flux and double stator based on mathematical modeling

The object of this research is electromechanical processes in a generator with an axial magnetic flux and a double stator and an additional non-contact excitation winding operating as part of autonomous electric power systems. The power of the additional excitation system is about 2 % of the generator power. The presence of an additional non-contact winding, which is powered by direct current, makes it possible to control the generator voltage by changing the excitation current. This resolves the task to stabilize the output voltage of the generator with permanent magnets when the load and shaft speed change. This paper reports the construction of a three-dimensional field axisymmetric mathematical model of the generator under study, which has made it possible to calculate and investigate its characteristics and parameters, in particular the magnitude of magnetic induction in all structural elements. The model built takes into account the influence of finite effects, magnetic scattering fields, and the radial-axial nature of the closure of the main magnetic flux and the magnetic flux of the additional excitation winding. The use of a structure with a double stator makes it possible to more efficiently utilize the usable volume of the generator and to increase its power. A mathematical model of the generator in the d-q coordinate system was built, which has made it possible to synthesize algorithms for controlling the automatic voltage stabilization system of the generator voltage under conditions of change in load and shaft speed. Control algorithms were developed on the basis of the concept of inverse dynamics problems in combination with minimizing local functionalities of instantaneous energy values, which ensures that the system is robust when changing generator parameters and that regulators are implemented in a simple way, due to the lack of differentiation operations. Based on the models built and algorithms developed, the quality of control of the generator's output voltage when the load and frequency of the generator change was investigated by modeling in the MATLAB/Simulink environment. When setting a jump in the rated load and changing the rotational speed within ±15 % of the rated value, the automatic stabilization system provides astatic voltage control at a given level of 48 V. The results can be practically used in the design of autonomous electric power systems with high energy conversion efficiency, in particular wind turbines and hydraulic units

UnderPressure: Deep Learning for Foot Contact Detection, Ground Reaction Force Estimation and Footskate Cleanup

AbstractHuman motion synthesis and editing are essential to many applications like video games, virtual reality, and film post‐production. However, they often introduce artefacts in motion capture data, which can be detrimental to the perceived realism. In particular, footskating is a frequent and disturbing artefact, which requires knowledge of foot contacts to be cleaned up. Current approaches to obtain foot contact labels rely either on unreliable threshold‐based heuristics or on tedious manual annotation. In this article, we address automatic foot contact label detection from motion capture data with a deep learning based method. To this end, we first publicly release Under Pressure, a novel motion capture database labelled with pressure insoles data serving as reliable knowledge of foot contact with the ground. Then, we design and train a deep neural network to estimate ground reaction forces exerted on the feet from motion data and then derive accurate foot contact labels. The evaluation of our model shows that we significantly outperform heuristic approaches based on height and velocity thresholds and that our approach is much more robust when applied on motion sequences suffering from perturbations like noise or footskate. We further propose a fully automatic workflow for footskate cleanup: foot contact labels are first derived from estimated ground reaction forces. Then, footskate is removed by solving foot constraints through an optimisation‐based inverse kinematics (IK) approach that ensures consistency with the estimated ground reaction forces. Beyond footskate cleanup, both the database and the method we propose could help to improve many approaches based on foot contact labels or ground reaction forces, including inverse dynamics problems like motion reconstruction and learning of deep motion models in motion synthesis or character animation. Our implementation, pre‐trained model as well as links to database can be found at github.com/InterDigitalInc/UnderPressure.

Transition methods for stochastic simulation of parametric uncertainty in inverse problems of orbital dynamics

Transition methods for stochastic simulation of parametric uncertainty in inverse problems of orbital dynamics

Construction of stochastic differential equations of motion in canonical variables

Galiullin proposed a classification of inverse problems of dynamics for the class of ordinary differential equations (ODE). Considered problem belongs to the first type of inverse problems of dynamics (of the three main types of inverse problems of dynamics): the main inverse problem under the additional assumption of the presence of random perturbations. In this paper Hamilton and Birkhoff equations are constructed according to the given properties of motion in the presence of random perturbations from the class of processes with independent increments. The obtained necessary and sufficient conditions for the solvability of the problem of constructing stochastic differential equations of both Hamiltonian and Birkhoffian structure by the given properties of motion are illustrated by the example of the motion of an artificial Earth satellite under the action of gravitational and aerodynamic forces.

Development of the control system for taking off the maximum power of an autonomous wind plant with a synchronous magnetoelectric generator

The object of this study is electromechanical processes in an autonomous wind power plant with a magnetoelectrical generator. Under actual conditions, the wind speed is constantly changing. The wind turbine works as efficiently as possible only at the rated value of wind speed. When the wind speed changes, the efficiency of converting mechanical wind energy into electrical energy decreases. Controlling the power of the electric generator when the wind speed changes is a relevant scientific and technical task. A maximum power selection control system based on the parameters of an experimental sample of a synchronous magnetoelectric generator has been designed and investigated. A feature of the synthesized control system is that it was developed on the basis of the concept of inverse dynamics problems in combination with minimization of local functionals of instantaneous energy values. The control law provides weak sensitivity to parametric perturbations of the object and carries out dynamic decomposition of the interdependent nonlinear system, which predetermines its practical implementation. Transient processes of the power, voltage, and current of the stator, as well as the voltage and excitation current were established when the wind speed changes from 3 to 8 m/s, as well as when the active electrical resistance of the load changes. The results of this study confirm the effectiveness of the maximum power take-off control system when wind speed and load change. When the wind speed changes within 3–8 m/s and the load by 50 %, the efficiency of converting mechanical wind energy into electrical energy increases by 15–40 % compared to the traditional magnetoelectric system. The findings of the current study are recommended for practical use in autonomous power plants based on wind turbines with generators with permanent magnets.

A superposition principle for the inhomogeneous continuity equation with Hellinger–Kantorovich-regular coefficients

We study measure-valued solutions of the inhomogeneous continuity equation where the coefficients v and g are of low regularity. A new superposition principle is proven for positive measure solutions and coefficients for which the recently-introduced dynamic Hellinger–Kantorovich energy is finite. This principle gives a decomposition of the solution into curves that satisfy the characteristic system in an appropriate sense. In particular, it provides a generalization of existing superposition principles to the low-regularity case of g where characteristics are not unique with respect to h. Two applications of this principle are presented. First, uniqueness of minimal total-variation solutions for the inhomogeneous continuity equation is obtained if characteristics are unique up to their possible vanishing time. Second, the extremal points of dynamic Hellinger–Kantorovich-type regularizers are characterized. Such regularizers arise, for example, in the context of dynamic inverse problems and dynamic optimal transport.

Evaluation of the uncertainty due to dynamic effects in linear measuring devices

The evaluation of the uncertainty due to systematic dynamic effects is addressed. When high dynamic performance is required, they should be compensated, by solving the associated inverse dynamic problem. When instead they are considered compatible with the target uncertainty, they may be simply included in the uncertainty budget. Furthermore, even in the case of dynamic compensation, a residual uncertainty remains, due to the imperfect compensation, and should be evaluated. Therefore, simple formulas are presented here, applicable to many classes of dynamic phenomena, including periodic, harmonic, transitory impulsive and stochastic stationary ones.

ОБ ОДНОЙ ОБРАТНОЙ ДИНАМИЧЕСКОЙ ЗАДАЧЕ ПОРОУПРУГОСТИ ДЛЯ ПОРИСТОЙ СРЕДЫ

An inverse dynamic problem of poroelasticity of piecewise-smooth shear coefficient with respect to additional information about vibrations of free surface points is considered. The Gupill hypothesis of equal propagation time of perturbations through the layers of a porous medium saturated with liquid is assumed fulfilled. Recursive formulas for recovering the unknown shift coefficient are obtained.

INVESTIGATE THE INFLUENCE OF THE WELDING SEAMS PROFILE ON DRIVING ENERGY FOR INDUSTRIAL ROBOTS

This paper focuses on investigating the influence of the welding seams in the working space on the driving energy for the six degrees of freedom FD-V8 industrial welding robot. The system of kinematics equations is built on the basis of multi-bodies mechanics theory and industrial robotics theory. The AGV algorithm is applied to solve the inverse kinematics problem (IKP) for each specific welding seam. The Lagrange-Euler method is used to build the differential equations of the robot motion. The robot driving energy is determined on the basis of solving the inverse dynamics problem (IDP). The results of this study have important implications in designing an appropriate welding seams trajectory to reduce energy consumption, as a basis for step by step building the optimizing energy consumption and production costs problem meeting the challenge of the current energy shortage in modern industrial production.

Synthesis of Stabilizing Control of a Quadcopter Based on Linear Matrix Inequalities

The relevance of the work is due to the widespread introduction of unmanned aerial vehicles, including quadcopters, in various areas of both civil and military applications. A review of various methods of controlling quadcopters, considering their features as nonlinear objects of high dimensionality, is performed. The work is devoted to stabilizing a quadcopter on a complex trajectory defined by functional coordinate relationships in 3D space. A nonlinear dynamic model of the quadcopter in a coupled coordinate system is constructed. The quadcopter control is based on a combination of two control actions. When solving the inverse dynamics problem, program control provides motion along a given trajectory. Stabilization of motion along the desired trajectory is provided by phase coordinate feedback. The stabilizing regulator ratios are found by the modal control method based on the solution of a linear matrix inequality using a linearized model. The found feedback ratios provide the required degree of stability of the closed-loop system, ensuring the quadcopter robustness to parametric perturbations. The legitimacy of this approach to the synthesis of control of a nonlinear system is substantiated by the topological equivalence theorem for the nonlinear system and the linearized model in that the nonlinear system has stable or unstable manifolds, which are analogs of the stable or unstable spaces of the linearized system. The results of computational experiments to estimate the error in reproducing a given quadcopter trajectory are presented. A simulation of the quadcopter behavior was performed, and the trajectory reproduction error was calculated to confirm the effectiveness of the synthesized stabilizing control and the optimal control based on the Pontryagin maximum principle. According to this criterion, the stabilizing control synthesized based on linear matrix inequalities is more effective for the quadcopter. Computational experiments were performed using the MATLAB application software package.

Influence of the initial velocity of the projectile on the kinematics of the projectile

The work uses a mathematical model for determining the frontal air resistance to the movement of the projectile, which based on solving the inverse problem of dynamics. Knowing the flight range at certain angles of aiming given in the firing tables, the functional dependence of the force of the frontal air resistance to the movement of the projectile on its speed and other factors is determined, taking into account the effect on the projectile of its weight and Coriolis force.
 It has been established that, depending on the aiming angle, the projectile speed during flight can combine stages of movement with supersonic,transonic, subsonic, decreasing or increasing speeds. Therefore, the functional dependence of the frontal drag force is determined for each stage separately.
 The influence of the initial velocity and the deviation from the nominal value for the OF-462ZH projectile fired from a 122-mm D-30 howitzer, charge is investigated. Depending on the aiming angle, the projectile speed during flight can combine stages of movement with supersonic,transonic, subsonic, decreasing or increasing speeds.
 It was found that the displacement values of the coordinates of the zeroing point of the trajectory of the projectile, determined on the basis of the proposed mathematical model, are almost the same in modulus when the initial velocity deviates by +∆V0; the tabular values of the displacements of the trajectory zeroing point, in general, are large in magnitude from the theoretical; the greatest discrepancy between them is achieved at an aiming angle of 45o00’; determination of which of the displacement values, theoretical or tabular, are real, is possible during field tests.

Model of the supporting leg of an anthropomorphous robot or exoskeleton with two movable links taking into account the dynamics of the electric drive. Power engineering: research, equipment, technology

Currently, the direction associated with the development of exoskeletons and anthropomorphic robots is experiencing rapid growth due to the increase in the computing power of microprocessors and the breakthrough development of the theory of control of complex systems, including electromechanical systems that simulate the biomechanics of the human musculoskeletal system. This paper presents a controlled mechatronic robotic model of the support leg of an anthropomorphic robot or exoskeleton with two moving links.GOAL. Mathematical modeling of the dynamics of the supporting leg of an exoskeleton or an anthropomorphic mechanism in the form of two moving links.METHODS. The main difference between the model presented in this study and those created earlier is the use of angles counted between links corresponding to the case of real operation of electric drives. To achieve the goal of the work, the methods of robotics, mathematical modeling, mechatronics, theoretical mechanics, the study of systems of ordinary differential equations, control theory, empirical data for the human musculoskeletal system were applied.RESULTS. For the model of the mechanism, a system of Lagrange equations of the second kind is written, direct and inverse problems of dynamics are solved for a given program control of the motion of a mechatronic robotic system. The results are presented graphically and as an animated visualization of the movement of the links. Calculations were carried out both without taking into account the dynamics of electric drives, and taking into account the rotation of the rotors of electric motors. It has been established that the influence of the dynamics of the rotor of the electric motor on the mechanism is significant.CONCLUSION. The developed methods for setting the program movement of the supporting leg of an exoskeleton or an anthropomorphic robot made it possible to solve direct and inverse problems of dynamics and establish the need to take into account the rotating rotor of an electric motor.