Force Error Research Articles

An inverse problem of recovering the heat source coefficient in a fourth-order time-fractional pseudo-parabolic equation

In this paper, we have considered the problem of recovering the time dependent source term for the fourth order time fractional pseudoparabolic equation theoretically and numerically, for the first time, by considering an additional measurement at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of physics and mechanics. The existence of unique solution to the problem has been discussed by means of the contraction principle and Banach Fixed-point theorem on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output force), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the forcing term. The proposed problem is discretized using the quintic B-spline (QnB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine lsqnonlin tool has been used to expedite the numerical computations. Both analytical and perturbed data are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.

Nodal force error and its reduction for material point methods

This work shows that with the presence of a prestress, the discretized momentum equation using the material point method (MPM) or the dual domain material point (DDMP) method is inconsistent with the continuum version of the equation. This inconsistency leads to incorrect wave speed predictions in prestressed materials and numerical instabilities. The source of this inconsistency is the nodal force calculation using the material points.To address this issue, we introduce an auxiliary stress field, which can be regarded as a local background stress. We then reformulate the nodal force calculation, such that it only depends on the local stress variation. The effect of the prestress is then subtracted away in the nodal force calculation, and the consistencies of MPM and DDMP are then restored.This reformulation only requires simple modifications to the original nodal force in both MPM and DDMP. After implementing this reformulated nodal force in a numerical code and comparing the results with manufactured solutions, we found significant improvements not only in the accuracy of the solutions, but also in the numerical stability for both MPM and DDMP. With the reformulated nodal force, the DDMP results converge rapidly as the number of material points increases.

A framework for nonlinear peak response evaluation from linear response spectra of base-isolated structures using the bilinear hysteretic model

This paper proposes a simple computational framework that can account for both properties of the base-isolated structure using the bilinear hysteretic model and reliably evaluate the nonlinear peak response without nonlinear response history analysis. The proposed computational framework requires the construction of new non-proportional damping models that match the base-isolated structure to accurately decompose the nonlinear multi-degree-of-freedom structure into multiple single-degree oscillators with intrinsic frequencies and damping ratios by equivalent linearization and complex-mode superposition methods. Each oscillator is excited by a response spectrum matching its own damping ratio, and this procedure is iteratively computed until the displacement error of the isolation layer achieves a preset value. Simultaneously, the peak nonlinear response of interest is calculated using the complex-mode-superposition complete square combination rule. The effect of peak response on nonlinear parameters is analyzed using a typical numerical example, and the relative error values for the applicable range are presented. Parametric research demonstrates that the new non-proportional damping matrix accurately reflects the modal characteristics of the base-isolated structure and prevents Rayleigh damping from suppressing the first-order modal characteristics. The particular example evaluation shows that the computational framework can effectively evaluate the nonlinear peak response of the BIS, with the isolation layer displacement error controlled within +5 %, the bottom shear force error within +20 %, and the top acceleration error between −10 % and +25 %.

An incremental singular value decomposition approach for large-scale spatially parallel & distributed but temporally serial data – applied to technical flows

The article presents a strategy and its algorithm to compile a simulation-accompanying, incremental Singular Value Decomposition (SVD) for time-evolving, spatially parallel discrete data sets. The framework addresses state-of-the-art PDE solvers for computational science and engineering applications. An important characteristic of such applications is that the spatial size of the data is often time-invariant and significantly exceeds the temporal size due to the large computational grid in 3D applications. Typical examples, which are also considered in this article, relate to results extracted from unsteady flow simulations. Herein, the flow data, which progresses over time, is frequently calculated spatially parallel based on domain decomposition strategies, which allow to parallelize the simulation on distributed memory machines following a Single Instruction Multiple Data (SIMD) concept.With a view to the memory-efficient reuse of (compressed) simulation results and their CPU time-saving, sufficiently accurate generation, the paper scrutinizes the efficiency of incremental/parallel SVD approaches for such simulation examples. To improve the computational efficiency, the introduction of a bunch matrix is proposed, which enables the aggregation of multiple time steps and SVD updates, and significantly increases the efficiency. The suggested strategy is verified and validated by simple 2D laminar single-phase flows and subsequently applied to more complex 2D and 3D turbulent two-phase flows. Emphasis is given to (a) the accuracy of SVD-based reconstruction, (b) the physical realizability of the reconstructed quantities, (c) the independence of domain partitioning, (d) an efficient snapshot bunching, and (e) related implementation aspects. In addition, the influence of lower and (adaptive) upper rank thresholds on the effort and accuracy is evaluated. A final application renders the practical benefits of the approach and refers to a merchant ship in head waves at Re = 1.4×107 and Fn = 0.26. The simulation involves 2880 processor cores and the related full-rank snapshot matrix has (108×104) entries. With a numerical overhead of O(10%), this snapshot matrix can be incrementally generated and compressed by O(95%). The compression is accompanied by only small errors in the integral force and local wave elevation of O(10−2%). This qualifies the method for an efficient subsequent data processing.

Improvement on the axial force accuracy of the ANCF Euler-Bernoulli beam element with the second-order approximate function of the centerline and precise constraint equations

The conventional Euler-Bernoulli beam element of the absolute nodal coordinate formulation (ANCF) yields incorrect axial force due to the existence of the first-order approximate function of the centerline and over-constraint equations. This work aims to improve the axial force accuracy of the element by using the second-order centerline approximate function and precise constraint equations. The second-order function includes the quintic Hermite shape function and the vector of nodal coordinates containing zeroth to second-order derivatives of position. The elastic force vector of the element and its tangent stiffness were then formulated based on the decoupled elastic line method. The exact kinematic requirements of three types of constraint points were fully clarified. On this basis, the corresponding concise constraint equations, constraint Jacobian and Hessian matrices were rigorously derived in the context of ANCF, where G2 continuity was demonstrated to be satisfied at the node acted by an inclined concentrated force, the relative rotation constraint of the rigid joint was described using the dot product of the gradients on both sides of the joint, and the fixing gradient direction at the partially-clamped end was expressed by using the zero cross product of the end gradients in the initial and deformed configurations. Large deformation problems of typical beams under fixed and follower loads were solved using five types of elements, where the results of the ABAQUS B22 elements were taken as a reference for validation. It is found that when using the first- and second-order beam elements having over-constraint equations of the three types of constraint points, obvious axial force error occurs near the corresponding constraint points. In comparison, a few second-order elements containing the derived constraint equations output correct axial force consistent with ABAQUS. However, a large number of the first-order elements containing the correct constraint equations still yield inaccurate axial force in certain beam regions.

Feedback hybrid force and position control of an upper limb exoskeleton to support human movement

In the paper, a feedback hybrid control including a force feedback control and a position control is proposed to control a four degree of freedom (4-dof) upper limb exoskeleton for supporting human movement at the shoulder, elbow and wrist joints. The novelty of the paper is that it has been able to control all the interaction forces at all links in the exoskeleton robot by using the proposed control. The desired interaction forces at the links and desired position are compared with the measured interaction forces and position, respectively. Then the torque at the shoulder, the torque elbow and the torque wrist joints are controlled to compensate the force error and the position error. The gains of the proposed controller are optimized by using the Balancing Composite Motion Optimization (BCMO). The simulation and control of the 4-dof upper limb exoskeleton using the proposed control is carried out in the paper to show that the interaction forces and the position of the exoskeleton track their desired values.

Motion-Control Strategy for a Heavy-Duty Transport Hexapod Robot on Rugged Agricultural Terrains

Legged agricultural transportation robots are efficient tools that can autonomously transport goods over agricultural terrain, and their introduction helps to improve the efficiency and quality of agricultural production. Their effectiveness depends on their adaptability to different environmental conditions, which is especially true for heavy-duty robots that exert ground forces. Therefore, this study proposes a motion-control strategy for a heavy-duty transport hexapod robot. Two critical tasks were accomplished in this paper: (1) estimating the support surface angle based on the robot’s foot position and body posture, and accordingly determining the motion constraint conditions on this support surface and the body posture based on energy optimization; (2) proposing an adaptive fuzzy impedance algorithm for real-time force–position composite control for adjusting foot position, in order to reduce the steady-state force tracking error caused by terrain stiffness, thus ensuring body stability through tracking of variable foot-end forces. An element of hardware in the loop control platform for a 3.55-ton device was designed and compared with the current popular force-control methods under different external contact terrains. The results show that the proposed control method can effectively reduce force errors, establish support forces faster on less-stiff environments, and reduce the torso tilt during phase switching.

Accuracy Assessment of Atomistic Neural Network Potentials: The Impact of Cutoff Radius and Message Passing.

Atomistic neural network potentials have achieved great success in accelerating atomistic simulations in complicated systems in recent years. They are typically based on the atomic decomposition of total properties, truncating the interatomic correlations to a local environment within a given cutoff radius. A more recently developed message passing (MP) neural network framework can, in principle, incorporate nonlocal effects through iteratively correlating some atoms outside the cutoff sphere with atoms inside, a process referred to as MP. However, how the model accuracy depends on the cutoff radius and the MP process has rarely been discussed. In this work, we investigate this dependence using a recursively embedded atom neural network method that possesses both local and MP features, in two representative systems: liquid H2O and solid Al2O3. We focus on how these settings influence predictions for structural and vibrational properties, namely, radial distribution functions (RDFs) and vibrational density of states (VDOSs). We find that while MP lowers test errors of energy and forces in general, it may not improve the prediction for RDFs and/or VDOSs if direct interatomic correlations in the local environment are insufficiently described. A cutoff radius exceeding the first neighbor shell is necessary, beyond which involving MP quickly enhances the model accuracy until convergence. This is a potentially more efficient way to increase the model accuracy than directly increasing the cutoff radius, especially with more memory savings in the GPU implementation. Our findings also suggest that using the mean test error as the measure of the model accuracy alone is inadequate.

Impaired Lumbar Extensor Force Control Is Associated with Increased Lifting Knee Velocity in People with Chronic Low-Back Pain

The ability of the lumbar extensor muscles to accurately control static and dynamic forces is important during daily activities such as lifting. Lumbar extensor force control is impaired in low-back pain patients and may therefore explain the variances in lifting kinematics. Thirty-three chronic low-back pain participants were instructed to lift weight using a self-selected technique. Participants also performed an isometric lumbar extension task where they increased and decreased their lumbar extensor force output to match a variable target force within 20–50% lumbar extensor maximal voluntary contraction. Lifting trunk and lower limb range of motion and angular velocity variables derived from phase plane analysis in all planes were calculated. Lumbar extensor force control was analyzed by calculating the Root-Mean-Square Error (RMSE) between the participants’ force and the target force during the increasing (RMSEA), decreasing (RMSED) force portions and for the overall force error (RMSET) of the test. The relationship between lifting kinematics and RMSE variables was analyzed using multiple linear regression. Knee angular velocity in the sagittal and coronal planes were positively associated with RMSEA (R2 = 0.10, β = 0.35, p = 0.046 and R2 = 0.21, β = 0.48, p = 0.004, respectively). Impaired lumbar extensor force control is associated with increased multiplanar knee movement velocity during lifting. The study findings suggest a potential relationship between lumbar and lower limb neuromuscular function in people with chronic low-back pain.

Research on Bowden Cable–Fabric Force Transfer System Based on Force/Displacement Compensation and Impedance Control

Bowden cable–fabric is a key force transfer device for flexible exoskeletons, and its precise control of force/displacement is a significant factor in the human–machine interaction of flexible exoskeletons. In this paper, a force/displacement control method based on friction compensation and impedance control was proposed based on a flexible Bowden cable–fabric force transfer testbed system. First, a set of in vitro experimental platforms simulating Bowden cable–fabric force transfer was built according to a typical flexible exoskeleton force transfer system, and following the walking gait of lower limbs, the expected force and knee joint motion were set. Secondly, the Bowden cable–fabric force transfer friction model was constructed as the basis of the system’s force transfer compensation. In addition, the stiffness model of Bowden cable–fabric and the lower leg movement model were established and combined with impedance control to realize the precise control of system displacement. Finally, the damping and stiffness parameters suitable for the system were obtained through the impedance control simulation. In terms of the experiment, an in vitro Bowden cable–fabric force transfer experimental platform was built, and the expected force with the input peak value of 40 N, 50 N, and 60 N was set. Through the friction and position compensation model of Bowden cable–fabric force transfer and impedance control, the relative root-mean-square errors of the output force and expected force were obtained as 2.53%, 2.16%, and 2.07%, respectively. Therefore, the effectiveness of the proposed method is verified, which provides a foundation for the engineering application of flexible exoskeletons.

Dual PID Adaptive Variable Impedance Constant Force Control for Grinding Robot

High-precision and low-overshoot force control are important to guarantee the material removal rate and surface quality of robot grinding. However, traditional force control methods are subjected to positional disturbance, stiffness disturbance, contact process nonlinearity, and force-position coupling, leading to difficulties in robot constant force control. Therefore, how to achieve smooth, stable, and high-precision constant force control is an urgent problem. To address this problem, a dual PID adaptive variable impedance control is established (DPAVIC). Firstly, PD control is used to compensate for the force error, and PID is used to update the damping parameters to compensate for the disturbance. Secondly, a nonlinear tracking differentiator is used to smooth the desired force and reduce the contact force overshoot. Then, the stability, convergence, and effectiveness of the force control algorithm are verified via theoretical analysis, simulations, and experiments. The force tracking error and overshoot of a conventional impedance controller (CIC), adaptive variable impedance control (AVIC), and DPAVIC are analyzed. Finally, the algorithm is used in grinding experiments on a thin-walled workpiece. The force tracking error is controlled within ±0.2 N, and the surface roughness of the workpiece is improved to Ra 0.218 μm.

Error Influence Simulation of the 500 m Aperture Spherical Radio Telescope Cable-Net Structure Based on Random Combinations

The reflector of a Chinese 500 m aperture spherical radio telescope is supported by a giant cable-net structure. In the actual operation process, active displacement observation is realized by connecting the actuators with the control cables to adjust the cable net, which requires high manufacturing and installation accuracy. In this study, an error sensitivity computing method based on a normal distribution is adopted to perform single-error computing and multi-error coupling computing and to investigate the effect of the length error of all the cables, tensioning force error of active surface cables, and installation error of external nodes on the cable force. The results show that the length error of the surface cables and the installation error of the external nodes are the main factors affecting the cable force, while the length error of the control cables is a secondary factor. The coupling effect of multiple errors is not the linear superposition of each error’s influence; therefore, all the error factors should be comprehensively considered for coupling computing to determine the control index. Through multi-error coupling computing, it is determined that the length error limits of the surface cables and control cables are ±1.5 mm and ±20 mm, respectively, the tensioning force error limit of the active surface cables is ±10%, and the installation error limit of the external nodes is ±50 mm.

Learning compliant dynamical system from human demonstrations for stable force control in unknown environments

Force control in unknown environments poses a significant challenge in robotics due to the necessity of adjusting impedance to accommodate environmental uncertainties. One potential solution is to transfer human variable impedance skills for delicate force control to robots. However, due to the short duration and small displacement of the contact transients, it is difficult to collect demonstration data and learn the control strategy used by human. This paper focuses on learning a stable force control policy from human demonstration during contact transients. We first propose a human-in-the-loop method for transferring human variable impedance skills during contact transient in unknown environments. Then, we analyze human demonstration data and conclude two observations: (1) Demonstrators adjust the end-effector velocity of the robot according to the force error, such that the velocity magnitude reduces nonlinearly as the force error decreases; (2) Demonstrators modify the slope of the velocity–force error curve according to environmental stiffnesses, such that the slope magnitude decreases as environmental stiffness increases. Finally, based on these observations, we proposed a novel human-inspired force control strategy called compliant dynamical system (CDS), which requires only actual contact force and rough estimation of environmental stiffness to achieve desired force control in contact tasks. The stability of the proposed CDS is rigorously proved using Lyapunov stability theory to guarantee accurate force control performance. The effectiveness of the proposed method is validated by simulation and real-world experiments involving single-arm contact with unknown environments and dual-arm cooperative transportation of an object.

Development of Particle Resolved - Subgrid Corrected Simulations: Hydrodynamic force calculation and flow sub-resolution corrections

Particle Resolved - Direct Numerical Simulation (PR-DNS) of fluid–solid particles is conducted to study hydrodynamic interactions at the interface between phases. An original distinction is proposed between Particle Resolved Simulation (PRS) and Particle Resolved - Direct Numerical Simulation (PR-DNS). In PR-DNS, velocity gradients and pressure are fully resolved, which required several dozens of meshes in the diameter of the particles (40 in Stokes configuration), whereas for PRS, with grid resolution of 10 meshes per diameter, the gradients are only partly resolved, resulting in a sub-resolution of hydrodynamic forces exerted by the fluid to the interface of solid particles. The very high computational cost, associated to PR-DNS, limits its application to academic cases, even though, it was originally developed to study the collective physical effects of a fluid–particle assembly. It explains why numerous simulations with a grid resolution of only a dozen meshes in the particle’s diameter – typically between 10 and 16 – referred to here as PRS, can be found in the literature. In these cases, the velocity gradients and pressure are not accurately computed, inducing an error in the hydrodynamic forces exerted by the fluid to the interface of particles. This paper aims to clarify the hydrodynamic interactions at this scale for a single particle settling in an infinite medium under Stokes configuration. For this purpose, an original method, designed for solid particles, is proposed to compute hydrodynamic forces. A mesh-dependent correlation is then built to numerically correct the partially resolved hydrodynamic forces obtained with PRS, in what we call Particle Resolved - Subgrid Corrected Simulations (PR-SCS). Finally, the correction is assessed for various Reynolds numbers and density ratios under the Stokes assumption.

Predictions of thumb, hand, and arm muscle parameters derived using force measurements of varying complexity and neural networks

Subject-specific musculoskeletal models are a promising avenue for personalized healthcare. However, current methods for producing personalized models require dense, biomechanical datasets that include expensive and time-consuming physiological measurements. For personalized models to be clinically useful, we must be able to rapidly generate models from simple, easy to collect data. In this context, the objective of this paper is to evaluate if and how simple data, namely height/weight and pinch force data, can be used to achieve model personalization via machine learning. Using simulated lateral pinch force measurements from a synthetic population of 40,000 randomly generated subjects, we train neural networks to estimate four Hill-type muscle model parameters and bone density. We compare parameter estimates to the true parameters of 10,000 additional synthetic subjects. We also generate new personalized models using the parameter estimates and perform new lateral pinch simulations to compare predicted forces using these personalized models to those generated using a baseline model. We demonstrate that increasing force measurement complexity reduces the root-mean-square error in the majority of parameter estimates. Additionally, musculoskeletal models using neural network-based parameter estimates provide up to an 80% reduction in absolute error in simulated forces when compared to a generic model. Thus, easily obtained force measurements may be suitable for personalizing models of the thumb, although extending the method to more tasks and models involving other joints likely requires additional measurements.

Exploring model complexity in machine learned potentials for simulated properties

Machine learning (ML) enables the development of interatomic potentials with the accuracy of first principles methods while retaining the speed and parallel efficiency of empirical potentials. While ML potentials traditionally use atom-centered descriptors as inputs, different models such as linear regression and neural networks map descriptors to atomic energies and forces. This begs the question: what is the improvement in accuracy due to model complexity irrespective of descriptors? We curate three datasets to investigate this question in terms of ab initio energy and force errors: (1) solid and liquid silicon, (2) gallium nitride, and (3) the superionic conductor Li_{10}Ge(PS_{6})_{2} (LGPS). We further investigate how these errors affect simulated properties and verify if the improvement in fitting errors corresponds to measurable improvement in property prediction. By assessing different models, we observe correlations between fitting quantity (e.g. atomic force) error and simulated property error with respect to ab initio values.Graphical abstract

Effect of Corner Radius of Square Thin-Walled Structures on Crashworthiness Indicators

Generally, the crash box on automobile vehicles is a thin-walled structure with a square cross-section. The majority of research was carried out for a long time to find the optimum crashworthiness indicator. In this study, numerical simulations and experimental tests are used to investigate the effect of the corner radius of a square cross-section thin-walled structure on crashworthiness indicators. Quasi-static analysis with mild steel material produces the mean force (Pm) error value is less than 3% while varying the corner radius ranging from zero to 1 mm, 2 mm, and 3 mm shows energy absorption (EA) and peak force (Pmax) decreased.

American Society of Biomechanics Journal of Biomechanics Award 2022: Computer models do not accurately predict human muscle passive muscle force and fiber length: Evaluating subject-specific modeling impact on musculoskeletal model predictions

Musculoskeletal models are valuable for studying and understanding the human body in a variety of clinical applications that include surgical planning, injury prevention, and prosthetic design. Subject-specific models have proven to be more accurate and useful compared to generic models. Nevertheless, it is important to validate all models when possible. To this end, gracilis muscle–tendon parameters were directly measured intraoperatively and used to test model predictions. The aim of this study was to evaluate the benefits and limitations of systematically incorporating subject-specific variables into muscle models used to predict passive force and fiber length. The results showed that incorporating subject-specific values generally reduced errors, although significant errors still existed. Optimization of the modeling parameter “tendon slack length” was explored in two cases: minimizing fiber length error and minimizing passive force error. The results showed that using all subject-specific values yielded the most favorable outcome in both models and optimization cases. However, the trade-off between fiber length error and passive force error will depend on the specific circumstances and research objectives due to significant individual errors. Notably, individual fiber length and passive force errors were as high as 20% and 37% respectively. Finally, the modeling parameter “tendon slack length” did not correlate with any real-world anatomical length.

Nonlinear electromagnetic force analysis and compensation control of electromagnetic linear actuator

Nonlinear electromagnetic force is a common phenomenon in electromagnetic linear actuators (EMLA). This nonlinearity limits the application in high-precision control systems. For facilitating the controller design, the influencing factors of nonlinear electromagnetic force were analyzed, and the quadratic polynomial with unknown weights was designed to approximate the nonlinear relationship among electromagnetic force, excitation current and displacement. An adaptive integral robust control algorithm based on electromagnetic force compensation (AIRC-FC) was designed, which combined the adaptive control law of electromagnetic force nonlinear compensation, stable feedback and error signal continuous integral robust control. The tracking performance and the adaptability of the EMLA with and without compensation control were analyzed under different loads. The results show that AIRC-FC improve the tracking performance of the EMLA effectively, and maintain high control accuracy under different loads.

Damage formation and evolution mechanisms in drilling CFRP with prefabricated delamination defects: Simulation and experimentation

Delamination is a common defect in carbon fiber reinforced polymer (CFRP) during the molding process. Accurately understanding the formation and evolution mechanisms of drilling damage in CFRP with delamination defects is helpful for effectively suppressing drilling defects with delamination. In this study, a macro-micro finite element model (FEM) of CFRP with prefabricated delamination defects during drilling is established to simulate and analyze the damage formation and evolution behavior. The circular polytetrafluoroethylene (PTFE) films are employed to simulate the delamination defects in CFRP. These samples with different sizes of prefabricated delamination defects are prepared for the drilling experiments. The results show that the maximum errors of the thrust force and the exit damage factor between the simulation and experiments are 8.75% and 4.2% respectively, indicating good reliability of the simulation model. Under the condition of CFRP with delamination defects, as the size of the delamination defects increases, the maximum thrust force decreases, while the maximum delamination propagation length and exit damage increase. Compared to the case without delamination defects, when the delamination diameter is 7 mm, the delamination propagation percentage and the maximum stress in CFRP increase by 52.17% and 155.4% respectively, and the maximum increment of the exit delamination damage factor is 38%.