Early-stage rehabilitation therapy for post-stroke patients consists of intensive and accurate training sessions. During these sessions, the therapist moves the patient’s joint within its range of motion repetitively. Patients, at this stage, often cannot control their muscles, and neurological disorders may occur and lead to undesirable movements. Thus, the therapist should train the joint gently to handle any sudden involuntary movements. Otherwise, the joint may undergo excessive torques, which may injure it. In this paper, we address this case and develop a clinical rehabilitation robotic system for training the knee joint taking into account the occurrence of these undesirable movements. The developed system has an innovative mechanism to measure interaction torques exerted by involuntary movements. Then, we introduce a new control approach consisting of an admittance controller and a proportional-derivative controller augmented by a radial basis function (PD-RBF) neural network. The PD-RBF guides the robot joint along a predefined trajectory, while the admittance part tracks any sudden interaction torques and updates the predefined trajectory accordingly. Thus, the robot trains the knee joint and once an undesirable movement occurs the robot gets along with this movement smoothly, then it gets back to the predefined trajectory. To validate the performance of the proposed admittance PD-RBF controller, we consider two controllers, an admittance adaptive sliding mode control and an admittance conventional PD one. Then, a compatarive study is conducted on these controllers via real-world experiments. The obtained results verify the efficiency of the admittance PD-RBF and prove its superiority over the other aforementioned controllers.

Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

A first attempt is made to use recently developed, non-conventional Artificial Neural Network (ANN) models with Multilayer Perceptron (MLP), Radial Basis Function (RBF) and Adaptive Neuro-Fuzzy Interference System (ANFIS) architectures to predict the fuel flow rate of a commercial aircraft using real data obtained from Flight Data Records (FDRs) of the cruise, climb and descent phases. The training of the architectures with a single hidden layer is performed by utilising the Delta-Bar-Delta (DBD), Conjugate Gradient (CG) and Quickprop (QP) algorithms. The optimum network topologies are sought by varying the number of processing elements in the hidden layer of the networks using a trial-and-error method. An evaluation of the approximate fuel intake values against the ideal fuel intake data from the FDRs indicates a good fit for all three ANN models. Thus, more accurate fuel intake estimations can be obtained by applying the RBF-ANN model during the climb and descent flight stages, whereas the MLP-ANN model is more effective for the cruise phase. The best accuracy obtained in terms of the linear correlation coefficient is 0.99988, 0.91946 and 0.95252 for the climb, cruise and descent phase, respectively.

Recent interest in describing the spatial distribution of weeds and studying their association with site properties has increased the use of interpolation to estimate weed seedling density from spatially referenced data. In addition, farmers and consultants adopting elements of site-specific farming practices are using interpolation methods for mapping weed densities as well as soil properties. This study was conducted to compare the performance of four interpolation methods, namely inverse-distance weighting (IDW), ordinary point kriging (OPK), minimum surface curvature (MC), and multiquadric radial basis function (MUL), with respect to their ability to map weed-seedling densities. These methods were evaluated on data from four weed species, velvetleaf, hemp dogbane, common sunflower, and foxtail species, of contrasting biology and infestation levels in corn and soybean production fields in Nebraska. Mean absolute difference (MAD) and root mean square (RMS) between the observed point sample data and the estimated weed seedling density surfaces were used to evaluate the performance of the interpolation methods. Four neighborhood search types were compared within each interpolation method, and Search3 (12 to 16 neighboring locations) generated an interpolated surface with the smallest MAD and RMS indicating the highest precision. IDW with a power coefficient of p = 4 gave the smallest MAD and RMS, as did a test with an elliptical search and no anisotropy. The level of precision of all four interpolation methods was very poor for weed species with low infestation levels (> 75% of field weed-free; MAD ranged from 100 to 187% of the observed mean density), whereas precision was improved for weed species with high infestation levels (< 25% of field weed-free; MAD ranged from 45 to 85%). No single interpolator appears to be more precise than another. Implications of this study indicate that grid sample spacing and quadrat size are more important than the specific interpolation method chosen.

In this paper, bymeans of a new recursive algorithm of non-tensor-product-typed divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimation is worked out, and an equivalence is obtained between high-order non-tensor-product-typed divided differences and high-order partial derivatives in the case of odd and even interpolating nodes, respectively. Thirdly, several numerical examples illustrate the recursive algorithms valid for the non-tensor-product-typed interpolating polynomials, and disclose that these polynomials change as the order of the interpolating nodes, although the node collection is invariant. Finally, from the aspect of computational complexity, the operation count with the bivariate polynomials presented is smaller than that with radial basis functions.

In order to overcome the possible singularity associated with the Point Interpolation Method (PIM), the Radial Point Interpolation Method (RPIM) was proposed by G. R. Liu. Radial basis functions (RBF) was used in RPIM as basis functions for interpolation. All these radial basis functions include shape parameters. The choice of these shape parameters has been and stays a problematic theme in RBF approximation and interpolation theory. The object of this study is to contribute to the analysis of how these shape parameters affect the accuracy of the radial PIM. The RPIM is studied based on the global Galerkin weak form performed using two integration technics: classical Gaussian integration and the strain smoothing integration scheme. The numerical performance of this method is tested on their behavior on curve fitting, and on three elastic mechanical problems with regular or irregular nodes distributions. A range of recommended shape parameters is obtained from the analysis of different error indexes and also the condition number of the matrix system. All resulting RPIM methods perform very well in term of numerical computation. The Smoothed Radial Point Interpolation Method (SRPIM) shows a higher accuracy, especially in a situation of distorted node scheme.

An application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) in solution of incompressible turbulent combined forced and natural convection is for the first time explored in the present paper. The turbulent flow equations are described by the low-Re number к – ε model with Launder and Sharma and Abe et al. closure coefficients. The involved temperature, velocity, pressure, turbulent kinetic energy and dissipation fields are represented on overlapping 5-noded sub-domains through the collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second order partial derivatives of the fields are calculated from the respective derivatives of the RBF’s. The involved equations are solved through the explicit time stepping. The pressure-velocity coupling is based on Chorin’s fractional step method. The adaptive upwinding technique, proposed by Lin and Atluri , is used because of the convection dominated situation. The solution procedure is represented for a 2D upward channel flow with differentially heated walls. The results have been assessed by achieving a reasonable agreement with the direct numerical simulation of Kasagi and Nishimura for Reynolds number 4494, based on the channel width, and Grash of number 9.6×105. The advantages of the represented mesh-free approach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforward applicability in non-uniform node arrangements.

Spherical radial basis functions are used to define approximate solutions to pseudodifferential equations of negative order on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the collocation method. A salient feature of our approach in this paper is a simple error analysis for the collocation method using the same argument as that for the Galerkin method.

In this paper we derive local error estimates for radial basis function interpolation on the unit sphere . More precisely, we consider radial basis function interpolation based on data on a (global or local) point set for functions in the Sobolev space with norm , where s>1. The zonal positive definite continuous kernel ϕ, which defines the radial basis function, is chosen such that its native space can be identified with . Under these assumptions we derive a local estimate for the uniform error on a spherical cap S(z;r): the radial basis function interpolant ΛXf of satisfies , where h=hX,S(z;r) is the local mesh norm of the point set X with respect to the spherical cap S(z;r). Our proof is intrinsic to the sphere, and makes use of the Videnskii inequality. A numerical test illustrates the theoretical result.

This paper presents an investigation into the development of parametric and non-parametric approaches for dynamic modelling of a flexible manipulator system. The least mean squares, recursive least squares and genetic algorithms are used to obtain linear parametric models of the system. Moreover, non-parametric models of the system are developed using a non-linear AutoRegressive process with eXogeneous input model structure with multi-layered perceptron and radial basis function neural networks. The system is in each case modelled from the input torque to hub-angle, hub-velocity and end-point acceleration outputs. The models are validated using several validation tests. Finally, a comparative assessment of the approaches used is presented and discussed in terms of accuracy, efficiency and estimation of the vibration modes of the system.

This is an extension of previous work which used an artificial neural network with a back-propagation algorithm and a lookup table to find the inverse kinematics for a manipulator arm moving along pre-defined trajectories. The work now described shows that the performance of this technique can be improved if the back-propagation is made to be adaptive. Also, further improvement is obtained by using the whole workspace to train the neural network rather than just a pre-defined path. For the inverse kinematics of the whole workspace, a comparison has also been done between the adaptive back-propagation algorithm and radial basis function.