Force Reconstruction Research Articles

Multi-parameter multiplicative regularization: An application to force reconstruction problems

This paper introduces a multi-parameter multiplicative regularization for force reconstruction problems. This approach allows exploiting the local prior information available on the sources to identify, while determining the related regularization parameters in an elegant and efficient manner. The aim of this paper is to assess the applicability of a multi-parameter regularization strategy compared to a single parameter formulation for reconstructing the external sources acting on a mechanical structure. A particular attention is also paid to the practical resolution of the regularization problem by implementing an original Iteratively Reweighted algorithm derived from the direct application of the first-order optimality condition. The performance of the proposed algorithm in terms of solution accuracy is compared with a more classical implementation based on an Iteratively Reweighted Least-Squares procedure. The interest of the proposed multi-parameter strategy is assessed numerically. Obtained results demonstrates that consistent reconstructions are obtained for high and moderate measurement noise level whatever the formulation considered (i.e. single or multi-parameter) provided that the suitable resolution algorithm is implemented. For very and extremely noisy input data, the single parameter strategy is more robust than the multi-parameter approach.

Single-sensor approach for impact localization and force reconstruction by using discriminating vibration modes

This study presents a novel strategy for identifying an impact event on a structure from vibration measurements. Compared to triangulation techniques that require at least three sensors on a plate-like structure, only one sensor is used in this work for localizing the impact point. The proposed approach consists in extracting specific modal ponderations as a signature of impact location. The force reconstruction problem is simultaneously addressed by fitting a parametric law. The proposed procedure captures the main load history parameters such as the impact time, its duration and its intensity. An impact test campaign is performed on a metallic plate, equipped with one accelerometer only, to confirm the validity of the proposed single-sensor approach. The methodology is efficient for accurately localizing impacts that are applied anywhere on the plate and for quickly estimating the main load history parameters.

Exploiting Adaptability in Soft Feet for Sensing Contact Forces

The large majority of legged robots currently employ ball-feet or flat-feet. More recently soft feet have been introduced, to improve walking performance on uneven grounds. Nevertheless, their novel adaptability requires sensor systems beyond traditional Force/Torque sensors to estimate the distribution of forces on the contact surface. This letter shows how a perception layer realized with Inertial Measurement Units allows a soft foot to reconstruct not only the shape of the foot - hinting at the shape of the ground beneath - but also, under precise hypotheses, the contact force distribution. The problem is theoretically formalized and analysed with a quasi-static approach in the Sagittal plane. Then, theoretical results are experimentally validated in a simplified foot-ground interaction scenario. The force reconstruction provided by the proposed method allows to correctly identify the sole contact location arising from obstacles with radius down to 1 cm.

Unknown impact force localisation and reconstruction in experimental plate structure using time-series analysis and pattern recognition

Impact force is the load that actual structures often experience, and its localisation and time-history reconstruction are important for structural health and safety monitoring. However, classical inversion techniques for random impact-force identification are time-consuming and infeasible; therefore, they cannot be utilized for engineering applications. This study adopted multiple time-series analyses and a pattern-recognition method to identify unknown random impact forces, and this strategy is known as the pattern-recognition method combined with a similarity metric (PRMCSM). Numerical simulations of the proposed method have been conducted; however, their findings have not been experimentally verified. From an experimental perspective, this study considered suspended steel plate as the research object, impact force as the load input, and acceleration as the response output. The PRMCSM algorithm based on cosine similarity was verified, and the expected results were obtained. Moreover, various similarity metric indices are proposed to implement the feature extraction of time-domain responses, and a novel index that is not lower than cosine similarity was finally determined and was verified based on the PRMCSM strategy. In summary, the method based on pattern recognition achieves the rapid identification of random impact forces and exhibits potential for practical engineering applications.

An iterative finite element-based method for solving inverse problems in traction force microscopy

Background and ObjectiveDuring the last years different model solutions were proposed for solving cell forces under different conditions. The solution relies on a deformation field that is obtained under cell relaxation with a chemical cocktail. Once the deformation field of the matrix is determined, cell forces can be computed by an inverse algorithm, given the mechanical properties of the matrix. Most of the Traction Force Microscopy (TFM) methods presented so far relied on a linear stress-strain response of the matrix. However, the mechanical response of some biopolymer networks, such as collagen gels is more complex. In this work, we present a numerical method for solving cell forces on non-linear materials. MethodsThe proposed method relies on solving the inverse problem based on an iterative optimization. The objective function is defined by least-square minimization of the difference between the target and the current computed deformed configuration of the cell, and the iterative formulation is based on the solution of several direct mechanical problems. The model presents a well-posed discretized inverse elasticity problem in the absence of regularization. The algorithm can be easily implemented in any kind of Finite Element (FE) code as a sequence of different standard FE analysis. ResultsTo illustrate the proposed iterative formulation we apply the theoretical model to some illustrative examples by using real experimental data of Normal Human Dermal Fibroblast cells (NHDF) migrating inside a 2 mg/ml collagen-based gel. Different examples of application have been simulated to test the inverse numerical model proposed and to investigate the effect of introducing the correct cell properties onto the obtained cell forces. The algorithm converges after a small number of iterations, generating errors of around 5% for the tractions field in the cell contour domain. The resulting maximum traction values increased by 11% as a consequence of doubling the mechanical properties of the cell domain. ConclusionsWith the results generated from computations we demonstrate the application of the algorithm and explain how the mechanical properties of both, the cell and the gel, domains are important for arriving to the correct results when using inverse traction force reconstruction algorithms, however, have only a minor effect on the resulting traction values.

Force reconstruction based on the union of singular spectral analysis and regularization method

To overcome the problem of some ill-posed inverse problem of force reconstruction, which is caused by the noise in the measured responses and small singular values of the structure, a technology of force reconstruction based on a hybrid method of singular spectral analysis (SSA) and the Landweber regularization method is proposed in this study for the first time. The SSA is used to filter the structural response before using Landweber regularization. A new choice method of phase space reconstruction dimension is theoretically proposed, and the minimum embedding dimension is determined by the concept of optimizing difference spectrum theory. The feasibility of this method was demonstrated through three kinds of force reconstructions. The numerical simulation results and an acoustic vibration experiment demonstrated that the proposed method is more effective than the traditional method.Communicated by Wei-Chau Xie.

Reconstruction of External Forces Beyond Measured Points Using a Modal Filtering Decomposition Approach

A methodology for the reconstruction of dynamic loads from vibration response data is developed which employs a well-correlated finite element model to permit load identification at locations where no response measurements are available; this is extended from a force reconstruction approach based upon modal filtering. Force reconstruction is approached from a modal perspective, where modal responses are estimated from physical response data, followed by independent reconstruction of modal forces for each modal oscillator. Modal force estimates are analyzed in aggregate via singular value decomposition to permit identification of input locations which are compatible with the estimated modal force patterns. Modal forces are then returned to the physical domain via transformation by modal matrices. Experimental validation is performed with a free-free beam subject to multiple impact.

A novel method for force identification based on filter and FEM

Force identification plays a vital role in vibration control and structural health monitoring. Force identification is an inverse problem to determine the excitation load from the responses. In general, the inverse problem has strong ill-posedness and ill-condition, especially in the case of low signal-to-noise ratio. Singular value decomposition and Tikhonov regularization are widely used for the reconstruction of the input force. Nevertheless, the anti-noise performance and accuracy of the two methods are still unsatisfactory. In this paper, we present a novel method for force identification based on a filter and the finite element method (FEM). The filter is used to suppress the noise of the input signals. FEM is introduced to reduce the discretization error and the regularization parameters are optimized by the genetic algorithm. After the signal is filtered, the characteristics of the signal will be changed. Especially, the initial and the last segments of the input signal will be lost. These changes may make the identified force inaccurate, even more inaccurate than the force identified with no filtering. To deal with this problem, cubic spline interpolation and a nonlinear autoregressive neural network are introduced to repair the input signals. Finally, three numerical examples and experiments are used to verify the superiority of the method.

Force Reconstruction for Uncertain Structure Based on Interval Model and Second-Order Perturbation Theory

In order to reconstruct the upper and lower bounds of dynamic excitations applied on the uncertain structure, an algorithm based on interval model and second-order perturbation theory is presented in this paper. First, interval model is built up by expressing the uncertain parameters of structure in interval form. Next according to second-order perturbation theory, structure characteristic matrices and input force vector are approximated as second-order Taylor polynomial expansion at the midpoint of uncertain parameters. After that the input force’s midpoint, first-order and second-order partial derivatives are respectively calculated by existing step-by-step integration method. Then addition and subtraction of the three components obtained in previous step are operated. Ultimately the upper and lower bounds of dynamic load can be identified. Numerical simulation results demonstrate this method is with the characteristic of high efficiency and precision. In addition, it is able to remain a relatively strong robustness under noise turbulence.

A hierarchical method for the impact force reconstruction in composite structures

Impact damage is a major concern for new generation aircraft composite components due to their low impact resistance capabilities. The development of an impact location and force reconstruction algorithm would provide rapid and efficient prediction of damage occurrence, thus making structures safer and creating maintenance inspection procedures more efficient, thus saving time and costs. However, state-of-the-art impact force reconstruction algorithms use reference data from numerical simulations and require a detailed knowledge of mechanical properties, which are difficult to obtain under real operational conditions. This paper presents a hierarchical impact force reconstruction algorithm that relies on experimental structural responses measured by a sparse array of surface bonded receiving ultrasonic transducers. This algorithm uses time reversal method to retrieve the location of an impact source and interpolation techniques based on hierarchical radial basis functions to calculate the transfer function at the impact point and reconstruct the impact force history. A number of impact testing were performed on a composite plate-like structure and a wing stringer-skin panel, and compared with impact force algorithms available in literature. Experimental results revealed that the proposed impact force reconstruction method was able to extrapolate the information associated with points far from the impact location and determine the impact force history with high level of accuracy in a real aircraft structure. Since the proposed algorithm requires the calibration of transfer functions from a very sparse training set of data and it does not need numerical models of the component under investigation, it demonstrates its potential as a useful monitoring tool for impact force reconstruction in composite components for full-scale aircraft structural applications leading to timely and cost-efficient inspections.

On a general Iteratively Reweighted algorithm for solving force reconstruction problems

The multiplicative ℓq-regularization has been recently introduced in structural dynamics for solving force reconstruction problems. Practically, the resolution of this regularization strategy requires the implementation of an iterative procedure. To this end, an Iteratively Reweighted Least-Squares algorithm has been originally implemented. The core idea of this algorithm is to replace the direct resolution of the inverse problem by an equivalent iterative procedure having an explicit and unique solution at each iteration. However, the exploitation of this very general idea allows defining other Iteratively Reweighted schemes. The present paper aims at comparing the overall performances of three particular Iteratively Reweighted algorithms for solving force reconstruction problems derived from a more general and original iterative procedure. The numerical applications proposed in this contribution highlight the ability of the considered algorithms in providing consistent regularized solutions with respect to various parameters such as the measurement noise level or the tolerance chosen to stop the iterative process.

A review of methods for excitation force reconstruction

1. Balageas D, Fritzen CP, Guemes A. Structural Health Monitoring. ISTE Ltd, London, UK, 2006. Google Scholar

On the interest of a space-time regularization for reconstructing sparse excitation sources

In time domain, force reconstruction problems are commonly solved from Tikhonov and LASSO regularizations. Practically, these approaches can lead to inaccurate reconstructions, if the sources to identify don’t share the same space-time characteristics or the corresponding force vector doesn’t exhibit the desired structure. To alleviate this potential drawback, we have recently introduced a multiplicative space-time regularization that allows exploiting one’s prior knowledge of the spatial distribution of the sources as well as their time history. In this contribution, the proposed regularization strategy is compared to the multiplicative Tikhonov and LASSO regularizations through an experimental application to point out the practical interest of exploiting simultaneously both spatial and temporal prior information in terms of quality and robustness of the reconstructed excitation sources.

A Hierarchical Impact Force Reconstruction Method for Aerospace Composites

This research work presents a hierarchical method able to reconstruct the time history of the impact force on a composite wing stringer-skin panel by using the structural responses measured by a set of surface bonded ultrasonic transducers. Time reversal method was used to identify the impact location by the knowledge of structural responses recorded from a set of excitation points arbitrarily chosen on the plane of the structure. Radial basis function interpolation approach was then used to calculate the transfer function at the impact point and reconstruct the impact force history. Experimental results showed the high level of accuracy of the proposed impact force reconstruction method for a number of low-velocity impact sources and energies.

Identification of external forces via truncated response sparse decomposition under unknown initial conditions

When structural dynamic design for smart control and health monitoring is executed, imposed excitations on structures are one of most important issues in structural engineering. Identification of structural excitations, such as force reconstruction, moving force identification, and so on, has drawn increasing attentions in the last decades. Assumption of known initial conditions is a precondition for existing deterministic force reconstruction methods. However, initial conditions are often unknown and hard to be estimated. To address this problem, a novel truncated response sparse decomposition method is proposed for calculating the external forces under unknown initial conditions. The truncated response sparse decomposition involves two basic steps, that is, response sparse decomposition and force estimation. First, a collection of basis vectors is defined for expressing unknown forces. Unknown initial conditions are represented in modal space. Structural responses induced by both external forces and initial conditions are then normalized as potential response features to form an image dictionary, which is adapted to decompose the measured responses via sparse regularization. Second, the response features with lower decomposed amplitudes and small scaling factors are eliminated from the decomposed results. The remaining features are used for estimating the external forces. In order to assess the accuracy and the feasibility of the proposed force reconstruction method, some numerical simulations on a planar truss structure and a series of experimental studies are carried out. The illustrated results show the robustness and the applicability of the proposed method for addressing the force reconstruction problem under unknown initial conditions. Some related issues are discussed as well.

A non-negative Bayesian learning method for impact force reconstruction

Detecting and identifying impact events, which may cause severe damages, is important in assessment of the integrity of many engineering structures. This paper presents a new approach for reconstructing the impact forces applied on engineering structures (e.g., composites) using sensor recordings of the structural responses. The problem is firstly formulated as a discretized deconvolution problem in the time domain with the impact force vector as the unknown. Then with consideration of the physical property of the impact force, an inverse analysis approach of Bayesian learning with non-negative regularization constraints is employed to solve the ill-posed deconvolution problem. The newly proposed impact force reconstruction approach is illustrated by experimental examples performed on a sandwich composite structure. Results have demonstrated the effectiveness and applicability of the proposed approach to reconstruct impact forces.

Quantitative reconstruction of time-varying 3D cell forces with traction force optical coherence microscopy

Cellular traction forces (CTFs) play an integral role in both physiological processes and disease, and are a topic of interest in mechanobiology. Traction force microscopy (TFM) is a family of methods used to quantify CTFs in a variety of settings. State-of-the-art 3D TFM methods typically rely on confocal fluorescence microscopy, which can impose limitations on acquisition speed, volumetric coverage, and temporal sampling or coverage. In this report, we present the first quantitative implementation of a new TFM technique: traction force optical coherence microscopy (TF-OCM). TF-OCM leverages the capabilities of optical coherence microscopy and computational adaptive optics (CAO) to enable the quantitative reconstruction of 3D CTFs in scattering media with minute-scale temporal sampling. We applied TF-OCM to quantify CTFs exerted by isolated NIH-3T3 fibroblasts embedded in Matrigel, with five-minute temporal sampling, using images spanning a 500 × 500 × 500 μm3 field-of-view. Due to the reliance of TF-OCM on computational imaging methods, we have provided extensive discussion of the equations, assumptions, and failure modes of these methods. By providing high-throughput, label-free, volumetric imaging in scattering media, TF-OCM is well-suited to the study of 3D CTF dynamics, and may prove advantageous for the study of large cell collectives, such as the spheroid models prevalent in mechanobiology.

17세기 濟州島의 육군 재건과 방어체계 변화

17세기 濟州島의 육군 재건과 방어체계 변화

An optimal Bayesian regularization for force reconstruction problems

In a paper, recently published in Mechanical Systems and Signal Processing, we have proposed a full Bayesian inference for reconstructing mechanical sources acting on a linear and time invariant structure. The main interest of this approach is to propose an estimation of all the parameters of the model and quantify the posterior uncertainty associated to each parameter. Since all the necessary information about the problem is available, statistical measures, such as the mean, the median and the mode of the solution, can be easily estimated. In many practical situations, however, one only wants to determine the most probable parameters given the available data. Consequently , it is not relevant to implement a full Bayesian inference to only extract a point estimate. To overcome this potential issue, this paper introduces an optimal Bayesian regularization aiming at computing the Maximum a Posteriori estimate of the Bayesian formulation previously introduced by the authors. In doing so, the most probable parameters are obtained without heavy computations. The validity of the proposed method is assessed numerically and experimentally. In particular, obtained results highlight the ability of the proposed regularization strategy in computing solutions with a minimal amount of prior information on the sources to identify.

High-performance reconstruction of microscopic force fields from Brownian trajectories

The accurate measurement of microscopic force fields is crucial in many branches of science and technology, from biophotonics and mechanobiology to microscopy and optomechanics. These forces are often probed by analysing their influence on the motion of Brownian particles. Here we introduce a powerful algorithm for microscopic force reconstruction via maximum-likelihood-estimator analysis (FORMA) to retrieve the force field acting on a Brownian particle from the analysis of its displacements. FORMA estimates accurately the conservative and non-conservative components of the force field with important advantages over established techniques, being parameter-free, requiring ten-fold less data and executing orders-of-magnitude faster. We demonstrate FORMA performance using optical tweezers, showing how, outperforming other available techniques, it can identify and characterise stable and unstable equilibrium points in generic force fields. Thanks to its high performance, FORMA can accelerate the development of microscopic and nanoscopic force transducers for physics, biology and engineering.