Nonlinear Least Research Articles

A generalized framework for identification and geometrical feature extraction of circular and polygonal shaped prisms

Abstract This paper introduces a versatile framework crucial for robotic applications such as object manipulation, mobile robot navigation, and pole climbing. It addresses the identification of geometric shapes and dimensions of diverse objects found in varied environments. The proposed method utilizes LiDAR scanning to capture objects from different angles, generating point clouds merged through transformations and superimpositions. After filtering and slicing, intersections are isolated and projected onto a chosen datum plane. The framework employs Non-Linear Least Square fitting via Gauss Newton iterative approach, utilizing pseudo-inverse Jacobian of a hypotrochoid to approximate polygons. The algorithm consecutively fits polygon prisms, determining the best fit with the least norm of error. Results indicate an average least square error of less than 9% for radius fitting and a high f-score for shape identification.

Nonlinear Stable Least Trimmed Squares for Regression Models with Stable Errors

The common method for estimating nonlinear regression coefficients with Gaussian errors is nonlinear least squares. It is not robust for regression with heavy-tailed errors. The nonlinear least trimmed squares method is an efficient procedure for regression with heavy-tailed errors. We are introducing nonlinear least trimmed squares for the nonlinear regression with non-Gaussian stable errors and estimate the parameters of the stable distributions based on the order statistics. The traditional least trimmed squares regression algorithm is adopted for errors with infinite variance by modifying the trimming procedure. This paper estimated the nonlinear regression coefficients with stable errors through the proposed nonlinear least trimmed squares algorithm based on a new trimming procedure based on the properties of stable order statistics moments. The proposed algorithm was applied to real and simulated datasets and evaluated using a bootstrap standard deviation estimator.

Finding Complex-Valued Solutions of Brent Equations Using Nonlinear Least Squares

Finding Complex-Valued Solutions of Brent Equations Using Nonlinear Least Squares

On the efficiency of misspecified Gaussian inference in nonlinear regression: Application to time-delay and Doppler estimation

Nonlinear regression plays a crucial role in various engineering applications. For the sake of mathematical tractability and ease of implementation, most of the existing inference procedures are derived under the assumption of independent and identically distributed (i.i.d.) Gaussian-distributed data. However, real-world situations often deviate from this assumption, with the true data generating process being a correlated, heavy-tailed and non-Gaussian one. The paper aims at providing the Misspecified Cramér–Rao Bound (MCRB) on the Mean Squared Error (MSE) of any unbiased (in a proper sense) estimator of the parameters of a nonlinear regression model derived under the i.i.d. Gaussian assumption in the place of the actual correlated, non-Gaussian data generating process. As a special case, the MCRB for an uncorrelated, i.i.d. Complex Elliptically Symmetric (CES) data generating process under Gaussian assumption is also provided. Consistency and asymptotic normality of the related Mismatched Maximum Likelihood Estimator (MMLE) will be discussed along with its connection with the Nonlinear Least Square Estimator (NLLSE) inherent to the nonlinear regression model. Finally, the derived theoretical findings will be applied in the well-known problem of time-delay and Doppler estimation for GNSS.

Mutual Inductance Estimation of SS-IPT System through Time-Domain Modeling and Nonlinear Least Squares

Mutual Inductance Estimation of SS-IPT System through Time-Domain Modeling and Nonlinear Least Squares

A stochastic approach to delays optimization for narrowband transmit beam pattern in medical ultrasound

In ultrasound imaging, state-of-the-art methods for transmit beam pattern (TBP) optimization have well-known drawbacks like non-uniform beam width over depth, significant side lobes, and quick energy drop-out beyond the focal region. To overcome these limitations, we develop a TBP optimization approach by focusing on its narrowband approximation and considering transmit delays as free variables instead of linked to specific focal depths. We formulate a non-linear Least Squares problem to minimize the difference between the TBP corresponding to a set of delays and the desired one, modeled as a 2D rectangle elongated along the beam axis. Three metrics are defined to quantitatively evaluate the results. Results obtained by synthetic simulations show that the main lobe width is considerably more intense (+31.5%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+31.5\\%$$\\end{document} on average) and uniform (+28.7%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+28.7\\%$$\\end{document} on average) over the whole depth range compared to classical focalized Beam Patterns. The application of the method to elastography shows improvements in the ultrasound energy concentration along a desired axis.

High Impedance Fault Detection Based on Arc Changing Tendency and Nonlinear Least Squares for Active Distribution Networks

High Impedance Fault Detection Based on Arc Changing Tendency and Nonlinear Least Squares for Active Distribution Networks

A Bayesian approach for exploring person × environment interaction within the environmental sensitivity meta-framework.

For investigating the individual-environment interplay and individual differences in response to environmental exposures as captured by models of environmental sensitivity including Diathesis-stress, Differential Susceptibility, and Vantage Sensitivity, over the last few years, a series of statistical guidelines have been proposed. However, available solutions suffer of computational problems especially relevant when sample size is not sufficiently large, a common condition in observational and clinical studies. In the current contribution, we propose a Bayesian solution for estimating interaction parameters via Monte Carlo Markov Chains (MCMC), adapting Widaman etal. (Psychological Methods, 17, 2012, 615) Nonlinear Least Squares (NLS) approach. Findings from an applied exemplification and a simulation study showed that with relatively big samples both MCMC and NLS estimates converged on the same results. Conversely, MCMC clearly outperformed NLS, resolving estimation problems and providing more accurate estimates, particularly with small samples and greater residual variance. As the body of research exploring the interplay between individual and environmental variables grows, enabling predictions regarding the form of interaction and the extent of effects, the Bayesian approach could emerge as a feasible and readily applicable solution to numerous computational challenges inherent in existing frequentist methods. This approach holds promise for enhancing the trustworthiness of research outcomes, thereby impacting clinical and applied understanding.

Effective optimization strategies for abrasive water jet machining of glass-carbon fiber reinforced composites: A comparative study of evolutionary optimization techniques

This study applied evolutionary optimization techniques and neural networks to predict optimum machining parameters of Abrasive Water Jet Machining (AWJM) for machining Glass-Carbon Fiber Reinforced Composite (GCFRC) materials. Several researchers have employed different optimization techniques; however, evolving computational capabilities further open avenues to optimize such parameters. Five evolutionary techniques, namely Artificial Neural Network (ANN), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Simulated Annealing (SA), and Nonlinear Least Square Error (LSE), were applied to minimize surface roughness (Ra) and maximize kerf width (Kw) and material removal rate (MRR). The output performance parameters (Ra, Kw, and MRR) were formulated as a linear mathematical function of machining parameters: tool feed rate (TFR), cutting speed rate (CSR), and stand-off distance (SOD). Though this study solves linear optimization problems, the proposed technique will be a strong tool for solving complex associations of machining and performance parameters in the future. A dataset of machining parameters and subsequent performance parameters was adopted from the available literature. The results indicated that the LSE method outperformed other techniques, yielding the lowest Root Mean Square Error (RMSE) in predicting Ra, Kw, and MRR, thus ensuring high machining accuracy. LSE technique reported relatively least RMSE values of 0.37 µm, 0.149Mm, and 237.23 mm3/min for Ra, Kw, and MRR, respectively. SA and PSO displayed identical and competitive RMSE values, slightly higher than LSE (up to 20 % higher). ANN and GA techniques were not effective relative to other considered techniques. LSE, SA, and PSO provide superior performance in optimizing AWJM parameters. The significant contribution of this research is the proposed optimization technique, offering a clear direction for solving complex associations between the performances and machining parameters of AWJM. This work also provides a foundation for future research to optimize such associations for other machining setups.

A Study of Correlative Modeling on Viscosity for Some Organic Binary Liquid Systems

In order to systematically investigate the correlating capability of viscosity models, six well-known correlative models of Grunberg-Nissan (GN), Hind (HND), Heric (HRC), Ausländer (AUS) for dynamic viscosity, η, and McAllister 3-body (MAC3) and McAllister 4-body (MAC4) for kinematic viscosity, ν, were employed and tested for viscosity data of 83 organic binary liquid systems consisting of 33 different aromatic hydrocarbons (ArH), alkanes (RH), cycloalkanes (CyRH) and alkanols (ROH). Keeping ArH as a common component and increasing the chain length of other components, the systems were categorized as Category 1: ArH + RH, Category 2: ArH + CyRH, Category 3: ArH + ArH and Category 4: ArH + ROH. For all the models fitting parameters along with the statistical parameters such as SPD σ(%), ASPD σ(%), OASPD σ(%) and GOASPD σ(%) were computed by the Nonlinear Least Squares Minimization (NLSM) technique with the ‘Solver’ add-in package. Among the four categories, Category 3, OASPD, σ(%) values lie between 0.16 and 0.21, indicating that all the models fitted extremely well. However, for dynamic viscosities, the η AUS model demonstrated the best correlating capabilities with GOASPD σ(%) of 0.48, while the HND model performed the poorest with GOASPD σ(%) at 3.8. For the kinematic viscosities, ν in both the MAC3 and MAC4 models yielded satisfactory results with GOASPD σ(%) as < 1.0 %.

A Comparison of R Tools for Nonlinear Least Squares Modeling

A Comparison of R Tools for Nonlinear Least Squares Modeling

Autonomous flight performance optimization of fixed-wing unmanned aerial vehicle with morphing wingtip

PurposeThe purpose of this paper is to improve autonomous flight performance of a fixed-wing unmanned aerial vehicle (UAV) via simultaneous morphing wingtip and control system design conducted with optimization, computational fluid dynamics (CFD) and machine learning approaches.Design/methodology/approachThe main wing of the UAV is redesigned with morphing wingtips capable of dihedral angle alteration by means of folding. Aircraft dynamic model is derived as equations depending only on wingtip dihedral angle via Nonlinear Least Squares regression machine learning algorithm. Data for the regression analyses are obtained by numerical (i.e. CFD) and analytical approaches. Simultaneous perturbation stochastic approximation (SPSA) is incorporated into the design process to determine the optimal wingtip dihedral angle and proportional-integral-derivative (PID) coefficients of the control system that maximizes autonomous flight performance. The performance is defined in terms of trajectory tracking quality parameters of rise time, settling time and overshoot. Obtained optimal design parameters are applied in flight simulations to test both longitudinal and lateral reference trajectory tracking.FindingsLongitudinal and lateral autonomous flight performances of the UAV are improved by redesigning the main wing with morphing wingtips and simultaneous estimation of PID coefficients and wingtip dihedral angle with SPSA optimization.Originality/valueThis paper originally discusses the simultaneous design of innovative morphing wingtip and UAV flight control system for autonomous flight performance improvement. The proposed simultaneous design idea is conducted with the SPSA optimization and a machine learning algorithm as a novel approach.

A Method for Solving Ill-Conditioned Nonlinear Least Squares Problems and Its Application in Image Distortion Correction Using Self-Calibration

In this study, the ill-conditioning of the iterative method for nonlinear models is discussed. Due to the effectiveness of ridge estimation for ill-conditioned problems and the lack of a combination of the H-K formula with the iterative method, the improvement of the LM algorithm is studied in this paper. Considering the LM algorithm for ill-conditioned nonlinear least squares, an improved LM algorithm based on the H-K formula is proposed for image distortion correction using self-calibration. Three finite difference methods are used to approximate the Jacobian matrix, and the H-K formula is used to calculate the damping factor in each iteration. The Brown model, quadratic polynomial model and Fourier model are applied to the self-calibration, and the improved LM algorithm is used to solve the model parameters. In the simulation experiment of space resection of a single image, we evaluate the performance of the LM algorithm based on the gain ratio (LMh) and the improved LM algorithm based on the H-K formula (LMHK), and the accuracy of different models and algorithms is compared. A ridge trace analysis is carried out on the damping factor to illustrate the effects of the improved algorithm in handling ill-conditioning. In the second experiment, the improved algorithm is applied to measure the diameter of a coin using a single camera. The experimental results show that the improved LM algorithm can reach the same or higher accuracy as the LMh algorithm, and it can weaken the ill-conditioning to a certain extent and enhance the stability of the solution. Meanwhile, the applicability of the improved LM algorithm in self-calibration is verified.

Attitude control of a 3-DoF quadrotor platform using a linear quadratic integral differential game approach

In this study, a linear quadratic integral differential game approach is applied to regulate and track the Euler angles for a quadrotor experimental platform using two players. One produces commands for each channel of the quadrotor and another generates the worst disturbance based on the mini-maximization of a quadratic criterion with integral action. For this purpose, first, the attitude dynamics of the platform are modeled and its parameters are identified based on the Nonlinear Least Squares Trust-Region Reflective method. The performance of the proposed controller is evaluated for regulation and tracking problems. The ability of the controller is also examined in the disturbance rejection. Moreover, the influence of uncertainty modeling is studied on the obtained results. Then, the performance of the proposed controller is compared with the classic Proportional Integral Derivative, Linear Quadratic Regulator, and Linear Quadratic Integral Regulator. The results demonstrate the effectiveness of the Game Theory on the Linear Quadratic Regulator approach when the input disturbance occurs.

Development of Deep Learning for Power Energy Optimization in the Industrial Robot System

This paper established power consumption modeling and motion estimation optimization of industrial robots. We also studied factors affecting the use of electrical energy, such as friction, torque, and electric current. The energy consumption parameters of each coupling can be quantified through the Deep Learning (DL) technique, Scaled Conjugate Gradient (SCG) estimation, or Simulation and experimentation based on the movement posture of a given robot dynamic model to control the robot operation. The robot dynamic model parameters can be identified and expressed in mathematical equations. Electrical energy consumption estimates were analyzed using the SCG technique to compare with the Nonlinear Least Squares (NLS) method using a large dataset of approximately 60,000 samples. The results showed accurate parameter prediction and electrical energy consumption estimation of the robot locomotion pose. The maximum errors in the SCG and NLS methods were 0.89% and 1.54%, respectively. It indicated that the electric energy consumption model using the SCG estimation method is more efficient than the NLS method.

A Hessian for Gaussian Mixture Likelihoods in Nonlinear Least Squares

A Hessian for Gaussian Mixture Likelihoods in Nonlinear Least Squares

Investigating the Growth of Bacteria using Double Sigmoid Model with Reparameterization

The growth of an organism can be modeled using a growth curve. However, bacteria's growth pattern differs from other organisms. Bacterial growth is divided into four phases: lag, logarithmic, stationary, and death. The experts re-parameterized the growth curve to match the growth phase of the bacteria. Bacterial growth patterns generally do not show a single sigmoid pattern but form two curves. Therefore, the double sigmoid model is more suitable. This study modeled the growth of the Pseudomonas putida bacteria by observing the optical density of the medium. Model parameters are estimated using the Non-Linear Least Square (NLS) method with the Gauss-Newton algorithm. The modeling results show that the double sigmoid model fits the growth curve of Pseudomonas putida better than the single sigmoid model. The Double Logistic model outperforms all models with the highest adjusted R2 and the smallest RMSE, AIC, and BIC values.

Using registry data to identify individual dairy cows with abnormal patterns in routinely recorded somatic cell counts

Data from the Danish milk recording system routinely enter the Danish Cattle Database, including somatic cell counts (SCC) for individual animals. Elevated SCC can signal intramammary inflammation, suggesting subclinical mastitis. Detecting mastitis is pivotal to limit severity, prevent pathogen spread, and target treatment or culling. This study aimed to differentiate normal and abnormal SCC patterns using recorded registry data.We used registry data from 2010 to 2020 for dairy cows in herds with 11 annual milk recordings. To create consistency across herds, we used data from 13,996 unique animals and eight different herds, selected based on the amount of data available, only selecting Holstein animals and conventional herds. We fitted log10-transformed SCC to days in milk (DIM) using the Wilmink and Wood’s curve functions, originally developed for milk yield over the lactation. We used Nonlinear Least Square and Nonlinear Mixed Effect models to fit the log10-transformed SCC observations to DIM at animal level.Using mean squared residuals (MSR), we found a consistently better fit using a Wood’s style function. Detection of MSR outliers in the model fitting process was used to identify animals with log10(SCC) curves deviating from the expected “normal” curve for that same animal. With this study, we propose a method to identify single animals with SCC patterns that indicate abnormalities, such as mastitis, based on registry data. This method could potentially lead to a registry data-based detection of mastitis cases in larger dairy herds.

AUTOMATED FINE-SCALE FOREST INVENTORY USING BACKPACK LIDAR – A STRATEGY BASED ON FEATURE EXTRACTION, MATCHING, AND TRACKING FROM INTEGRATED SCANS

Abstract. Backpack LiDAR systems are gaining popularity due to their rapid data acquisition, portability, and cost-effectiveness. However, using backpack LiDAR in forest poses challenges, such as GNSS signal attenuation under the canopy, leading to inaccurate trajectory estimates and misregistered point clouds. This paper aims at presenting a novel method that addresses this challenge by leveraging the integrated scans (IS) concept to enhance point cloud quality for automated forest inventory. The proposed method for forest inventory using ISs consists of four key steps: (i) Integrated scans (IS) generation, (ii) feature extraction, matching, and tracking, (iii) trajectory enhancement, and (iv) tree biometric extraction. Firstly, IS point clouds are generated based on the initial GNSS/INS trajectory. Secondly, reliable forest features such as tree trunks and ground patches are extracted, matched, and tracked across ISs. These features are then utilized in the trajectory enhancement step, where a non-linear Least Squares Adjustment (LSA) technique is used to minimize discrepancies among the features to enhance the trajectory. The resulting point clouds, based on the improved trajectory, are used to extract tree biometric information. The proposed method was evaluated using two distinct datasets collected with different systems. The evaluation results, both qualitatively and quantitatively, validate the effectiveness of the proposed method, showcasing its potential for fine-scale forest inventory applications.

Estimating Optimal Cost, Insulation Layer Thickness, and Structural Layer Thickness of Different Composite Insulation External Walls Using Computational Methods

All the modern gadgets and space conditioning in buildings consume lots of energy. Energy consumption can be optimized using Composite Insulation External Walls (CIEW) built from mortar plaster and structural and insulation layers. This study aimed to improve the overall performance of CIEW by optimizing the structural and insulation layer thickness. The objective was to minimize the Life Cycle Cost (LCC) and maximize the Life Cycle Savings (LCS) of CIEW. The nonlinear Least Squares Estimation (LSE) optimization technique for optimizing LCC and LCS of CIEW was used in the study. The study considered three insulation materials—Extruded Polystyrene (XPS), Rock Wool (RW), and Glass Wool (GW)—across three heat sources, including Circulating Fluidized Bed (CFB), Grate-Fired Boiler (GFB), and Air-Source Heat Pump (ASHP). The Life Cycle Cost Analysis (LCCA) methodology suggested by Huang using a traditional optimization technique was used as a basis for mathematical formulations and result comparison. The payback period of CIEW with optimal structural and insulation layer thickness was computed. The findings revealed that applying the LSE method enabled greater economic efficiency than the LCCA method, with an up to 9.12% increase in LCS value and an up to 7.41% decrease in LCC value. The research also revealed significant correlations between insulation and structural layer thicknesses and economic parameters.