Jacobian Estimation Research Articles

Target tracking using the efficient estimation of the image Jacobian with large residual

An efficient calibration-free visual servoing method is presented. The nonlinear optimization problem is formulated with the residual term for tracking a target. The residual term is estimated based on the secant approximation method. The image Jacobian is estimated using the affine model of the end-effector feature. Some experimental results are presented for a moving target.

APPROXIMATIONS WITH VORTICITY BOUNDS FOR THE GINZBURG–LANDAU FUNCTIONAL

We propose an approximation scheme for complex-valued functions defined on a smooth domain Ω: the approximating functions have a Ginzburg–Landau energy of the same magnitude as the initial function, but they possess moreover improved bounds on vorticity. As an application, we obtain a variant of a Jacobian estimate first established by Jerrard and Soner. This variant was conjectured by Bourgain, Brezis and Mironescu.

Efficient electromagnetic optimization of microwave filters and multiplexers using rational models

A method is presented for the efficient optimization of microwave filters and multiplexers designed from an ideal prototype. The method is based on the estimation of a rational function adjusted to a reduced number of samples of the microwave device response obtained either through electromagnetic analysis or measurements. From this rational function, a circuital network having the previously known topology of the microwave device is synthesized and compared to a circuital network with the desired response but including nonidealities. All of the process of analysis and model extraction can be seen as a model function that relates the physical parameters of the microwave device with the extracted circuital network parameters. Then, the error vector of the circuital parameters is used to generate a correction vector of the physical parameters through an estimation of the inverse of the Jacobian matrix of the complete model function. The Jacobian estimation is updated at each iteration of the optimization process with no need for additional evaluations of the model function. Two numerical examples of the proposed technique corresponding to the synthesis of a filter and a diplexer are presented, demonstrating the increased efficiency of the proposed technique with respect to direct electromagnetic optimization.

Distribution of the estimated lyapunov exponents from noisy chaotic time series

Abstract. In this paper, we give statistical analyses and simulation studies on the Lyapunov exponents estimated from noisy chaotic time series. Through the Jacobian estimation approach, the asymptotic distribution of the estimated Lyapunov exponents are studied and characterized from the observed noisy chaotic time series. Theoretical results are visualized and verified by numerical simulations.

On an open problem for Jacobians raised by Bourgain, Brezis and Mironescu

We establish a Jacobian estimate in the context of Ginzburg–Landau theory, which was conjectured in a recent work of Bourgain, Brezis and Mironescu. To cite this article: F. Bethuel et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Enhancing the estimation of plant Jacobian for adaptive neural inverse control

To implement the specialized learning of the inverse dynamic neuro-controller for controlling nonlinear plants with noise, it is strongly desirable that an on-line trained neural plant emluator may provide a reasonably good estimation of the plant Jacobian under noise environments. This paper presents an approach for enhancing the estimation of the plant Jacobian which is on-line used in direct adaptive neural inverse control schemes. The estimated teaching signals are obtained by using the input–output data available at each time step, and then they are used to train the neural plant emluator by a new cost function introduced in this work for training the plant emluator. Convergence theorem for the adaptive back-propagation algorithm and stability of the closed-loop control system are established by using the Lyapunov theory. Simulations are conducted for demonstrating the effectiveness of the proposed strategy.

Multiple camera model-based 3-D visual servo

A multiple camera, visual servo system is presented that tracks and aligns rigid objects in 3D using wire-frame models. The system is capable of achieving frame-rate (30 Hz between two cameras) object tracking while using nonlinear iterative pose estimation. Tracking begins with a rough estimate of the positions of the objects. The relative rotation and translation of the cameras (external calibration) is calculated from the images. Six degree-of-freedom object pose estimates are computationally fused to provide tracking that is both stable and robust to occlusion. The tracking data is used to move (servo) the objects together while performing online robot Jacobian estimation. The robot is controlled in real time by joint angles with no prior knowledge of its kinematics.

The role of exploratory movement in visual servoing without calibration

The calibration requirements for visual servoing can make it difficult to apply in many real-world situations. One approach to image-based visual servoing without calibration is to dynamically estimate the image Jacobian and use it as the basis for control. However, with the normal motion of the robot towards the goal, the estimation of the image Jacobian deteriorates over time. We propose the use of additional “exploratory motion” in the direction in which it is most needed, thus considerably improving the estimation of the image Jacobian. We study the role such exploratory motion can play in a visual servoing task. Simulations and experiments with a 6-DOF robot are used to verify the practical feasibility of the approach.

Statistical analysis of Lyapunov exponents from time series: A Jacobian approach

In this paper, we provide a statistical analysis for the Lyapunov exponents estimated from time series. Through the Jacobian estimation approach, the asymptotic distributions of the estimated Lyapunov exponents of discrete-time dynamical systems are studied and characterized based on the time series. Some new results under weak conditions are obtained. The theoretical results presented in the paper are illustrated by numerical simulations.

Visually guided object grasping

We present a visual serving approach to the problem of object grasping and more generally, to the problem of aligning an end-effector with an object. First, we extend the method proposed by Espiau et al. (1992) to the case of a camera which is not mounted onto the robot being controlled, and we stress the importance of the real-time estimation of the image Jacobian. Next, we show how to represent a grasp or more generally, an alignment between two solids in 3D projective space using an uncalibrated stereo rig. Such a 3D projective representation is view-invariant in the sense that it can be easily mapped into an image set-point without any knowledge about the camera parameters. Finally, we perform an analysis of the performances of the visual servoing algorithm and of the grasping precision that can be expected from this type of approach.

Optimal Estimation of Jacobian and Hessian Matrices That Arise in Finite Difference Calculations

In this paper, the problem of estimating Jacobian and Hessian matrices arising in the finite difference approximation of partial differential equations is considered. Using the notion of computational molecule or stencil, schemes are developed that require the minimum number of differences to estimate these matrices. A procedure applicable to more complicated structures is also given.

Optimal estimation of Jacobian and Hessian matrices that arise in finite difference calculations

In this paper, the problem of estimating Jacobian and Hessian matrices arising in the finite difference approximation of partial differential equations is considered. Using the notion of computational molecule or stencil, schemes are developed that require the minimum number of differences to estimate these matrices. A procedure applicable to more complicated structures is also given.