Mixed redundancy, which combines active and cold-standby redundancy, can improve reliability design flexibility but substantially increases computational complexity. Consequently, it is rarely used in multi-objective reliability-redundancy allocation problems (MRRAPs), as balancing conflicting objectives proves challenging. To address it, this paper formulates a bi-objective RRAP (BRRAP) with mixed redundancy, aiming to maximize system reliability while minimizing cost. The reliability of cold-standby and mixed redundant subsystems is precisely evaluated using continuous time Markov chain models. Although swarm intelligence algorithms are widely used for MRRAPs because of their global search capability and implementation simplicity, their stochastic updating mechanism often leads to weak local exploitation and premature convergence. To overcome this limitation, an importance measure (IM)-based two-stage BRRAP optimization framework is developed, which iteratively combines swarm intelligence-based global search with IM-guided local refinement. By adjusting Pareto solutions from both reliability and redundancy perspectives, the IM-based local optimization effectively pushes the Pareto front toward higher reliability and lower cost. Experiments based on four benchmarks demonstrate that the proposed framework improves solution quality, convergence, and diversity of Pareto fronts.

Contour accuracy is crucial for machining precision in five-axis computer numerical control (CNC) machining. This paper addresses the challenge of improving contour accuracy by proposing a novel contour error estimation and compensation method based on dynamic time warping (DTW). By incorporating time information and geometric characteristics of the machining path, the proposed method introduces a multiple-information fusion algorithm to define distance characteristics between the planned and actual trajectory sequences. This allows the calculation of a distortion path and the establishment of a mapping model between the two sequences. To mitigate the effect of DTW singularity on contour error estimation, a mapping model is established between line segments to determine reference points. The position contour error and the direction contour error of the five-axis tool are accurately estimated using segmented Hermite interpolation, and a spatial iterative learning framework is employed to compensate for them. Experimental results demonstrate the effectiveness of the proposed method in dealing with estimation errors in complex trajectories and its good performance in improving contour accuracy.Note to Practitioners—This paper proposes an effective strategy for the estimation of contour errors in five-axis machining. Currently, most methods for contour error estimation in five-axis machining are based on the nearest point principle. However, this approach fails to accurately estimate contour errors for complex trajectories with significant curvature variations, leading to ineffective contour error compensation in subsequent stages. Therefore, we introduce a contour error estimation algorithm based on DTW. This algorithm takes into account the time information and geometric features of the machining path. Experimental results validate the feasibility and advantages of this approach.

In radial basis function neural network (RBFNN)-based real-time learning tasks, forgetting mechanisms are widely used such that the neural network can keep its sensitivity to new data. However, with forgetting mechanisms, some useful knowledge will get lost simply because they are learned a long time ago, which we refer to as the passive knowledge forgetting phenomenon. To address this problem, this article proposes a real-time training method named selective memory recursive least squares (SMRLS) in which the classical forgetting mechanisms are recast into a memory mechanism. Different from the forgetting mechanism, which mainly evaluates the importance of samples according to the time when samples are collected, the memory mechanism evaluates the importance of samples through both temporal and spatial distribution of samples. With SMRLS, the input space of the RBFNN is evenly divided into a finite number of partitions, and a synthesized objective function is developed using synthesized samples from each partition. In addition to the current approximation error, the neural network also updates its weights according to the recorded data from the partition being visited. Compared with classical training methods including the forgetting factor recursive least squares (FFRLS) and stochastic gradient descent (SGD) methods, SMRLS achieves improved learning speed and generalization capability, which are demonstrated by corresponding simulation results.

Memory, as the basis of learning, determines the storage, update, and forgetting of knowledge and further determines the efficiency of learning. Featured with the mechanism of memory, a radial basis function neural network (RBFNN)-based learning control scheme named real-time progressive learning (RTPL) is proposed to learn the unknown dynamics of the system with guaranteed stability and closed-loop performance. Instead of the Lyapunov-based weight update law of conventional neural network learning control (NNLC), which mainly concentrates on stability and control performance, RTPL uses the selective memory recursive least squares (SMRLS) algorithm to update the weights of the neural network and achieves the following merits: 1) improved learning speed without filtering; 2) robustness to hyperparameter setting of neural networks; 3) good generalization ability, i.e., reuse of learned knowledge in different tasks; and 4) guaranteed learning performance under parameter perturbation. Moreover, RTPL realizes continuous accumulation of knowledge as a result of its reasonably allocated memory while NNLC may gradually forget knowledge that it has learned. Corresponding theoretical analysis and simulation studies demonstrate the effectiveness of RTPL.

When the persistent excitation (PE) condition is met, neural network control based on deterministic learning can approximate the true dynamics of nonlinear systems. However, in this approach, learning speed and accuracy are severely constrained by the PE level. In this article, we investigate the explicit relationship between the PE level and input signals. Specifically, this research investigates a neural network structure determined by the mechanical characteristics of a computer numerical control (CNC) machine tool. We explore a method to generate training trajectories that fill the designated feature space or repeatedly pass through hidden layer nodes, ensuring that deterministic learning achieves a sufficient PE level. Then, we validated the effectiveness of the proposed method through experiments conducted on a three-axis CNC machine tool using actual machining trajectories. The experimental results consistently confirmed that the generated training trajectories endow the RBF neural network with more feature information than random NURBS trajectories. Additionally, the tracking error and contour error of the CNC machine tool were significantly reduced.

Under the persistent excitation (PE) condition, the real dynamics of the nonlinear system can be obtained through the deterministic learning-based radial basis function neural network (RBFNN) control. However, in this scheme, the learning speed and accuracy are limited by the tradeoff between the PE levels and the approximation capabilities of the neural network (NN). Inspired by the frequency domain phase compensation of linear time-invariant (LTI) systems, this paper presents an adaptive phase compensator employing the pure time delay to improve the performance of the deterministic learning-based adaptive feedforward control with the reference input known a priori. When the adaptive phase compensation is applied to the hidden layer of the RBFNN, the nonlinear approximation capability of the RBFNN is effectively improved such that both the learning performance (learning speed and accuracy) and the control performance of the deterministic learning-based control scheme are improved. Theoretical analysis is conducted to prove the stability of the proposed learning control scheme for a class of systems which are affine in the control. Simulation studies demonstrate the effectiveness of the proposed phase compensation method.

In this paper, a contour error control strategy based on spatial iterative learning control (sILC) is developed for repetitive processing of five-axis computer numerical control (CNC) machine tools. A curve approximation method with an adaptive moving window is developed to achieve accurate tool position and orientation contour error estimation, and a five-axis contour error control strategy based on sILC is proposed. The compensation method is derived using an sILC algorithm to modify the geometric reference path instead of modifying the controller. The experimental results show that the proposed control strategy reduces contour errors of five-axis CNC machine tools, and it outperforms the traditional tracking error control. Note to Practitioners—This paper aims to propose an effective five-axis contour error control scheme. At present, most of the five-axis contour error control methods rely on the modification of the controller, which is not allowed in commercial CNC systems. Therefore, we propose a five-axis contour error compensation algorithm based on sILC, which modifies the system’s reference path. Experimental results show that the proposed method can reduce the five-axis contour errors, while maintaining the machining efficiency.

Abstract In this paper, we propose a hybrid offline/online neural networks learning method, which combines complementary advantages of two types of neural networks (NNs): deep NN (DNN) and single-layer radial basis function NN (RBFNN). Firstly, after analyzing the mechatronic system’s model, we select reasonable features as the input of the DNN to learn the inverse dynamic characteristics of the closed-loop system offline, so as to establish the mapping between the desired trajectory and the reference trajectory of the system. The trained DNN is used to generate a new reference trajectory and compensate for the tracking error in advance, which can speed up the convergence of online learning control based on RBFNN. This reference trajectory is further modified iteratively when the tracking task is repeated. For this purpose, a single-layer RBFNN model is established, and an online learning algorithm is developed to update the RBFNN parameters. The proposed hybrid offline/online NN method can improve the tracking performance of mechatronic systems by modifying the reference trajectory on top of the baseline controller without affecting the system stability. To verify the effectiveness of this method, we conduct experiments on a piezoelectric drive platform.

In typical contour-following applications such as computer numerical control (CNC) machining, contouring error characterizes the surface quality of final workpieces. The traditional task coordinate frame (TCF)-based approach transforms the contouring control problem into a 3-D regulation problem along the axes of the local Frenet frame. The contouring error is then controlled indirectly by two decoupled regulation systems. In order to achieve a satisfactory contouring error performance, two sets of control parameters must be tuned for the two systems, respectively. In this paper, a novel TCF (nTCF) is proposed. Given the nearest position from the actual position to the desired (or approximated) contour, one axis is set along the line passing the actual position and the nearest position and another axis is along the advancing direction. The system dynamics in the world Cartesian coordinate frame is transformed into nTCF. The contouring control problem for 3-D contours can be reduced and it locally becomes a 2-D regulation problem, one in contouring direction and the other in advancing direction. By implementing the computed-torque control, two proportional-derivative controllers are integrated to regulate the advancing error and the contouring error, respectively. The contouring error is directly regulated by tuning a single set of control parameter, which becomes much simpler than that in the TCF-based approach, where two sets of control parameters are needed to tune. Experiments on an industrial three-axis glass engraving computer numerical control (CNC) machine show the validity of the proposed nTCF-based approach with two typical 3-D contours. Note to Practitioners-In machining applications, the contouring error is a crucial index with respect to the surface quality of machined parts. In this paper, a novel task coordinate frame (nTCF) is proposed to deal with the 3-D contouring control issue. By transforming the system dynamics from the world Cartesian coordinate frame to nTCF in real-time control, a 3-D contouring control problem is locally reduced and transformed into a 2-D error dynamics regulation problem. In the nTCF, the contouring performance and advancing performance can be decoupled and separately regulated by only two parameters in 3-D contouring control applications. The contouring error can be directly controlled. Moreover, existing contouring error calculation (estimation) techniques, such as linear approximation, circular approximation, or analytical methods, can be readily integrated into the nTCF-based approach. Both analysis and experiments show that the nTCF-based approach is effective and much simpler in parameter tuning for 3-D contouring control.

How to achieve the required contouring tracking accuracy especially during high-speed and large-curvature contouring tasks, has always been an important problem in manufacturing applications. In this paper, a contouring error estimation method based on natural local approximation is used, and then the position-loop cross-coupled controller is proposed to reduce the estimated contouring error. The effectiveness and superiority of the natural local approximation method using on the position-loop cross-coupled control scheme are demonstrated through experiments on a biaxial linear motor drive servo system.

In view of the successful application of planar parallelogram in the Delta robot and its variants, we are interested to investigate mechanisms consisting of spatial parallelograms. The spatial parallelogram, denoted by Pscra*, is a 2-SS (S stands for a spherical joint) parallel mechanism having identical length for opposite links. We show that a 3-PPscra* mechanism is generically undergoes 3-dimensional purely translational motion. Based on the 3-PPscra* topology, an integrated optimal design on both architecture and geometry design is carried out. Using the formulation of maximizing effective cubic workspace, the Orthopod, which has three orthogonally arranged linear joint axes, is found to be the best in our settings. A prototype machine of the Orthopod is thus designed and manufactured.