Inverse Jacobian Research Articles

Symbolic and Numerical Mathematical Modeling of the Jacobian of the Yaskawa Motoman - Gp7 Robot using Matlab©

Robotic manipulators have besides static positioning problems, but besides static we have in the robot its speeds and forces applied in any kind of movement. one of the show important activities of robot motion control is the definition of its movements in the workspace. to perform these moves its joints need I ‘m moving and to define these positions in the spaces we need to calculate the direct kinematics and the opposite we call the inverse kinematics. The main objective of this paper is to present a study on velocity analysis, known as the Jacobian matrix. The methodology employed in this exploratory scientific research will ok developed from experimental tests, bibliographic references and case studies applied at the Advanced robotics Institute (IAR). the robot under study is the YASKAWA-MOTOMAN-GP7 present at the robotics Laboratory of the IAR. The work brings as contribution the implementation of the Denavit-Hartenberg notation and the validation of the Jacobian matrix of the robot. The results are the determination of the equations of the inverse kinematics using the method based on the inverse Jacobian matrix. The code was implemented in MATLAB© software capable of proving the developed mathematical model, which is applicable in industrial robots.

On the effect of nonlinearity and Jacobian initialization on the convergence of the generalized Broyden quasi‐Newton method

AbstractThis article investigates two aspects of the generalized Broyden quasi‐Newton method that have a major impact on its convergence: the initial approximation of the Jacobian and the presence of nonlinearities in the secant conditions. After reformulating the common representation of generalized Broyden, a straightforward interpretation is given. This leads to a natural extension of the method in which an application‐dependent physics‐based surrogate model is used as initial approximation of the (inverse) Jacobian. A carefully chosen surrogate has the potential to greatly reduce the required number of iterations. The behavior of generalized Broyden depends strongly on the parameter that determines how many secant conditions are satisfied by the Jacobian approximation. Respecting all secant conditions reduces it to Anderson acceleration; a single one to Broyden's original method. An analysis demonstrates that these two variants behave very differently when nonlinearities are present in the secant conditions: they are ignored by Broyden, but can destabilize Anderson. On the other hand, the analysis shows that Broyden tends to neglect small linear information, possibly reducing convergence speed. To mitigate stability problems with Anderson acceleration, a practical method to detect and remove nonlinear secant information is introduced next. Finally, we solve a steady free‐surface‐flow problem using several generalized Broyden variants, testing the influence of the surrogate, the nonlinearities and the combination thereof. The results agree with the theoretical predictions, showing large differences in convergence behavior. Furthermore, the proposed method effectively negates the problems related to nonlinearities in this case.

A simple algorithm for diffuse optical tomography without Jacobian inversion

A computationally simpler algorithm to reconstruct the optical property distribution of turbid media using diffuse optical tomographic principles is presented. The proposed algorithm eliminates the requirement of large Jacobian matrix inversion which otherwise is essential for tomographic imaging. The most significant Jacobians are identified based on proper thresholding of the measurement and the intersection of these Jacobians gives the approximate spatial location of the inhomogeneity. The algorithm is tested and optimized using simulations and further validated using tissue-mimicking phantom-based experiments and in-vivo small-animal experiments.

Probabilistic three-phase power flow in a distribution system applying the pseudo-inverse and cumulant method

Abstract A new, analytical approach using the cumulant method is proposed for the three-phase probabilistic power-flow (PPF) analysis. The approach to forming the sensitivity matrix is based on quantifying the pseudo-inverse instead of the inverse jacobian matrix, since it is commonly singular in a distribution power network. The results are compared with those obtained using the point-estimate method (PEM) and the Monte Carlo (MC) method, which is a commonly used reference method for the PPF analysis in a distribution power network.

Simulation of flow in deformable fractures using a quasi-Newton based partitioned coupling approach

We introduce a partitioned coupling approach for iterative coupling of flow processes in deformable fractures embedded in a poro-elastic medium that is enhanced by interface quasi-Newton (IQN) methods. In this scope, a unique computational decomposition into a fracture flow and a poro-elastic domain is developed, where communication and numerical coupling of the individual solvers are realized by consulting the open-source library preCICE. The underlying physical problem is introduced by a brief derivation of the governing equations and interface conditions of fracture flow and poro-elastic domain followed by a detailed discussion of the partitioned coupling scheme. We evaluate the proposed implementation and undertake a convergence study to compare a classical interface quasi-Newton inverse least-squares (IQN-ILS) with the more advanced interface quasi-Newton inverse multi-vector Jacobian (IQN-IMVJ) method. These coupling approaches are verified for an academic test case before the generality of the proposed strategy is demonstrated by simulations of two complex fracture networks. In contrast to the development of specific solvers, we promote the simplicity and computational efficiency of the proposed partitioned coupling approach using preCICE and FEniCS for parallel computations of hydro-mechanical processes in complex, three-dimensional fracture networks.

End-Effector Position Estimation and Control of a Flexible Interconnected Industrial Manipulator Using Machine Learning

The control of flexible robot manipulators is a challenging task, especially when one considers parallel and interconnected manipulators under flexibility considerations. This paper proposes a method to estimate the position of the end-effector of a flexible interconnected manipulator based on a virtual sensor principle and function approximation schemes. By using SolidWorks/MSC ADAMS software, we developed a virtual prototype of a flexible interconnected manipulator, and rigorously evaluated the feasibility of using function approximation schemes such as Neural Networks (NN), Support Vector Machines (SVM), and Gaussian Process (GP) in estimating the deflection error arising due to the flexibility of the robot structure. Our rigorous computational experiments have shown that: (1) the NN, SVM, and GP models were are able to attain the promising and reasonable prediction accuracy, (2) a feedforward NN with 535 neurons and an Ascending distribution of its nodes achieves the best prediction and generalization to unseen environments (the upper bound of the error was <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.15\times 10^{-3}$ </tex-math></inline-formula> m); implying the robust estimation of the position of the end-effector under flexibility considerations, and (3) the control based on the inverse Jacobian and a NN-based estimator was able to follow a sinusoidal trajectory with reasonable tracking and error performance in MSC ADAMS & MATLAB/Simulink co-simulation. Our results show the feasibility and effectiveness of the nonlinear relationships learned by NN, SVM, and GP in aiding estimation and control of the position of the end-effector of the flexible manipulator with a promising/desirable capability.

An adaptive moving grid finite difference method

The grid generation is very crucial for the accuracy of the numerical solution of PDEs, especially for problems with very rapid variations or sharp layers, such as shock waves, wing leading and trailing edges, regions of separation, and boundary layers. The adaptive grid generation is an iterative approach to accommodate these complex structures. In this paper, we introduce a deformation based adaptive grid generation method, in which a differentiable and invertible transformation from computational domain to physical domain is constructed such that the cell volume (Jacobian determinant) of the new grid is equal to a prescribed monitor function. A vector field is obtained by solving the div-curl system and can be used to move the grids to the desired locations. By computing the inverse of Jacobian, any deformed grids can also be transformed back to the uniform grid. Several numerical results in two dimensions are presented. Some applications in image registration are discussed.

Quadrilateral-area-coordinate-based numerical manifold method accommodating static and dynamic analysis

Quadrilateral mesh is a commonly used mathematical cover for numerical manifold method (NMM). However, the NMM based on quadrilateral isoparametric mapping has some intrinsic shortcomings, which is revealed for the first time in this study. Therefore, a new NMM is established to overcome these drawbacks by adopting a quadrilateral area coordinate system. In the proposed NMM, the stiffness matrix can be analytically determined without resorting to cumbersome Jacobian inversion and numerical integration. Moreover, a cone-complementary-based contact model is formulated in the context of the new NMM framework, which enables accurate determination of frictional and cohesive contact forces in solving dynamic contact problems. Thus, artificial penalty and open-close iteration in the original NMM can be all avoided. Several benchmark examples are simulated to demonstrate the excellent performance of the presented NMM.

Kinematic Analysis of a Parallel Manipulator Driven by Perpendicular Linear Actuators

In this paper, a goniometer-type specimen stage with a linear actuation mechanism mounted on a rotation mechanism is introduced. The linear actuation mechanism was modeled as a spatial parallel manipulator consisting of a moving body, three linear actuators, and an anti-rotation mechanism. The three linear actuators were arranged perpendicular to each other. In the specimen stage, the linear actuators were in ball contact with the surface of a holder designed to hold a specimen. For the parallel manipulator, the ball contact was replaced with two prismatic joints and a spherical joint. The mobility of the manipulator without the anti-rotation mechanism was one degree of freedom greater than the number of actuators. Therefore, the redundant one degree-of-freedom motion was restrained using an anti-rotation mechanism with three rotation joints and two prismatic joints. The inverse and direct kinematics of the goniometer mechanism were derived and verified. In addition, the inverse Jacobian was derived, and local and global performance indices were analyzed by the terms of manipulability and isotropy. Finally, the goniometer-type specimen stage was designed by the global performance indices.

MODIFICATION OF PSB QUASI-NEWTON UPDATE AND ITS GLOBAL CONVERGENCE FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS

Nonlinear problems mostly emanate from the work of engineers, physicists, mathematicians and many other scientists. A variety of iterative methods have been developed for solving large scale nonlinear systems of equations. A prominent method for solving such equations is the classical Newton’s method, but it has many shortcomings that include computing Jacobian inverse that sometimes fails. To overcome such drawbacks, an approximation with derivative free line is used on an existing method. The method uses PSB (Powell-Symmetric Broyden) update. The efficiency of the proposed method has been improved in terms of number of iteration and CPU time, hence the aim of this research. The preliminary numerical results show that the proposed method is practically efficient when applied on some benchmark problems.

Experimental and analytical evaluation of tool path error using computer integrated nonlinear kinematical modeling for a 4DOF parallel milling machine

ABSTRACT Machining errors in parallel kinematic machines primarily depend on their extent of kinematic nonlinearity. In this paper, a computer integrated model of the kinematic nonlinearity and trajectory interpolation method for a 4DOF parallel milling machine is developed. The nonlinear errors prompted during various trajectory modes are analyzed. It is proved that in order to avoid significant non-linearity, the actuator and the end-effector space must possess identical order of trajectory. The effects of tool path length, its spatial location, and the Jacobian matrix properties on the kinematic nonlinearity error, are studied. Results showed that the path length is the governing factor for the nonlinear behavior of the mechanism. Moreover, the kinematic error is illustrated to have a reverse relationship with maximum singular values of the inverse Jacobian matrix. Experiments are conducted using digital dial indicators. Experimental results verified the accuracy of the proposed mathematical approach and confirmed its applicability in actual machining processes. Moreover, the proposed interpolation algorithm successfully limits the kinematic error under the machining tolerance by minimal segmentation. Finally, it is demonstrated that the proposed method is superior in terms of accuracy, to the median osculating circle (MOC) method.

Application of parametric function in construction of particle shape and discrete element simulation

In previous studies, particles, such as ellipsoids and super-ellipsoids, are mainly described by implicit functions. However, parametric functions can define more surfaces that can be used to represent a wider variety of particle shapes. In this study, parametric functions are used to construct particles, and the algorithm to determine the multi-point contact between concave particles is also given. Furthermore, the geometric parameters at the contact point, for instance, normal vectors, curvatures, and overlapping, are formulated by the parameters in the parametric function. To verify the proposed method, we use the discrete element method to simulate the systems of torus-shaped particles nested within each other, and analyze the momentum and kinetic energy changes with time. The equilibrium state with approximately zero energy is obtained, which means that the algorithm for multi-point contact of concave particles is suitable and stable. In addition to the torus-shaped particle, another two concave particles defined by parametric function are modeled. The simulation results indicate that the method is universal. Any plane curve can be used to construct surfaces by using the method provided in this work, and computational efficiency of simulations of particles defined by parametric function is higher because calculations of the inverse Jacobian matrix in the Newton-Raphson method are unnecessary. The parametric function method extends the scope of previous studies on particle shape.

Adaptive control of parallel robots with uncertain kinematics and dynamics

One of the most challenging issues in adaptive control of robot manipulators with kinematic uncertainties is the requirement of inverse Jacobian matrix in regressor form. This requirement is inevitable in the case of the control of parallel robots, whose dynamics formulation are derived in the task space. In this paper, an adaptive controller is proposed for parallel robots based on representation of Jacobian matrix in regressor form with asymptotic trajectory tracking. The main idea of this paper is separation of determinant and adjugate of Jacobian matrix in order to represent them into a new regressor forms. Simulation and experimental results on a 2-DOF RPR and a 3-DOF redundant cable driven robot, respectively, verify promising performance of the proposed method in practice.

Neural-dynamics-enabled Jacobian inversion for model-based kinematic control of multi-section continuum manipulators

Continuum manipulators are a new generation of robotic systems that possess infinite number of degrees of freedom associated with inherent compliance, unlike traditional robotic manipulators which consist of a finite number of rigid links. Because of this characteristic, controlling continuum manipulators is more complicated and difficult based on only traditional control theory. Soft computing techniques are solid alternative for improving the control performance of such kinds of robots. In this paper, we employ two types of neural dynamic approaches, i.e., gradient neural dynamics and zeroing neural dynamics, to solve the real-time Jacobian matrix pseudo-inversion problem, thereby achieving model-based kinematic control of multi-section continuum manipulators. Different kinds of neural dynamic models are investigated and their performances in terms of tracking accuracy are shown with and without noise disturbances. Simulation validations with a two-section and a three-section continuum manipulator demonstrate the feasibility and robustness of the proposed models.

A New Algorithm for Calibration of an Omni-Directional Wheeled Mobile Robot Based on Effective Kinematic Parameters Estimation

Calibration is a strategy to compensate for the systematic errors of a mobile robot hence, to increase the accuracy of the robot localization. So far, various methodologies have been proposed for the calibration of wheeled robots, but the majority have focused on non-holonomic mobile robots, e.g., differential type, and holonomic ones have been less studied from this point of view. This paper presents an innovative approach for the calibration of a holonomic robot by introducing “Effective Kinematic Parameters” (EKPs). To estimate the EKPs of a holonomic mobile robot, some tests are proposed, and the variables of the inverse Jacobian equations are modified by defining and minimizing an appropriate cost function. In the following, a virtual holonomic robot is modeled with some unknown intentional errors in its parameters, which cause some error in its path tracking. Then, the methodology is applied to estimate its EKPs, and the simulation is fulfilled again. The results show incredible improvement in path tracking. The methodology is applied to a 3-wheeled Omni-directional mobile robot, and some experiments are performed in a laboratory environment to experimentally evaluate the approach. The outcomes approved the algorithm and showed great enhancement in the calibration of the robot compared to the previous researches. The methodology can be employed to calibrate other robotic systems as well.

Kinematics Analysis of A Parallel Robot with 3 DOF and 4 Segments in Pure Translation.(Dept.M)

In this paper we present the direct and inverse analysis of a geometrical model of an UPS manipulator with three degrees of freedom added to a PPP passive central segment. This structure provides a pure translation motion. We will also determine the relations between generalized and articular velocities by using the inverse Jacobian matrix, Further, we determine the reciprocal relations between cartesian and angular velocities of the end effector via articular velocities by direct re-inversion of the Jacobian matrix. This study aims of implementing the control of a parallel robot manipulator. A prototype of a parallel robot has been built up in our laboratory in order to validate the proposed models.

Self-Tuning Fuzzy Task Space Controller for Puma 560 Robot

Since in most robot applications the desired paths are determined in task space or Cartesian space, it is important to control the robot arm in task space. In this paper, a fuzzy controller with modifiable scaling factors is proposed to control the robot end-effector in task space. The controller is a fuzzy system with a mechanism to change the scaling factors when the error is bounded under a predetermined value. The controller is designed in joint space and is developed to work space by using inverse Jacobian strategy. The simulations results on Puma 560 robot manipulator illustrate the high performance of the presented control method.

Inverse Jacobian Adaptive Tracking Control of Robot Manipulators with Kinematic, Dynamic, and Actuator Uncertainties

In this paper, we mainly solve the adaptive control problem of robot manipulators with uncertain kinematics, dynamics, and actuators parameters, which has been a long-standing, yet unsolved problem in the robotics field, because of the technical difficulties in handling highly coupled effect between control torque and the mentioned uncertainties. To overcome the difficulties, we propose a new Lyapunov-based adaptive control methodology, which effectively fuses the inverse Jacobian technique and the actuator adaptation law, with which the chattering in tracking errors caused by actuator parameter perturbation is well suppressed. It is demonstrated that the asymptotic convergence of all closed-loop signals is guaranteed. Moreover, the effectiveness of our control scheme is illustrated through simulation studies.

On the inverse problems of the polynomial vector fields in ℝ3

In this paper, we deduce the normal forms of polynomial differential systems in ℝ3 having, Fermat cubic surface, Clebsch surface, Goursat surface, Racor surface and 2-dimensional tours as invariant objects. We prove if we have an odd number of polynomial differential equations with the determined inverse Jacobian multipliers then we can make a new polynomial differential system of the first systems so that it’s inverse Jacobian multiplier can be determined by the inverse Jacobian multipliers of the first systems. If new system which is made by them has a limit cycle then we determine it’s position.

Real-time static voltage stability assessment in large-scale power systems based on spectrum estimation of phasor measurement unit data

Real-time static voltage stability assessment is crucial to a stressed modern grid. This paper, based on the moment-based spectrum estimation method, proposes a data-driven approach to gain insight into changes of voltage magnitudes via synchronous phasor measurement unit (PMU) data, such that proper actions can be taken against voltage collapse. Built upon random matrix theory, an inverse Jacobian related indicator, i.e., spectrum estimation based stability indicator (SESI), is introduced in this paper to track the motion of the system. After considering the relationship between moments and distributions, this method allows for the regime where the sample size is significantly smaller than the dimensionality. It is adaptive to the real-time static voltage stability assessment in large-scale power systems where the sample size in a sliding window is extremely smaller than the dimensionality of measurement variables, hence delivering a recent status with a low computational cost. In this way, the static voltage stability assessment of a power system with massive amounts of measurement variables can be more sensitively and efficiently implemented. Case studies with the IEEE 118-bus system, the IEEE 300-bus system, and a Polish 2383-bus system verify the effectiveness of the proposed approach.