Water quality prediction provides timely insights for addressing potential water environmental issues. Transformer-based models have been widely used in water quality prediction. However, the following challenges exist: 1) Noise in the time series of water quality causes nonlinear models to be overfit; 2) It is difficult to identify temporal correlations in complex time series data; and 3) Information utilization is limited in long-term prediction. This work introduces a large-scale water quality prediction model named SVD-Autoformer to address them. SVD-Autoformer combines aSavitzky-Golay (SG) filter,variational modedecomposition (VMD), anauto-correlation mechanism, and a deep decomposition architecture, which is achieved in the renovation of the transformer. First, the SG filter removes noise while retaining valuable data features. SVD-Autoformer employs the SG filter as a data preprocessing tool to reduce noise and prevent nonlinear models from overfitting. Second, VMD extracts major modes of the signals and their respective center frequencies, thus providing richer features for the prediction. Third, the deep decomposition architecture with embedded decomposition modules allows for gradual decomposition during the prediction process. SVD-Autoformer employs the architecture to extract more predictable components from complicated water quality time series for long-term forecasting. Finally, SVD-Autoformer applies the auto-correlation mechanism to capture the temporal dependence and enhance information utilization. Numerous experiments are conducted and the results demonstrate that SVD-Autoformer provides superior prediction accuracy over other advanced prediction methods with real-world datasets. Note to Practitioners—This paper explores the critical aspects of time series water quality prediction, aiming to provide valuable insights for engineers and decision-makers. Traditional water quality prediction methods primarily rely on linear time series approaches and suffer from high computational complexity when dealing with large-scale data. This study is motivated by the transformer architecture with highly parallel computing capability and innovatively proposes deep decomposition architecture to extract more predictable components. In practice, to handle massive data with low time complexity, we introduce an auto-correlation mechanism. We conduct experiments using real-world datasets to demonstrate that this method achieves superior water quality prediction accuracy. Additionally, the method has been deployed in a real-world water quality prediction platform. Our future work includes its applications to different real-world datasets arising from electric power, intelligent transportation, and meteorological rainfall prediction.

Water quality prediction provides timely management to solve possible water environmental problems, which is of great importance. However, the following challenges exist: 1) The existence of noise in the water quality time series can lead to overfitting of nonlinear models; 2) It is difficult to capture temporal dependencies in complex time series data; 3) Long-term forecasting is difficult to achieve. To address the above difficulties, this work proposes a multi-step water quality prediction model, called SG-Autoformer, which combines the Savitzky-Golay filter, the inner series decomposition, and an auto-correlation mechanism. First, SG-Autoformer performs noise reduction on the water quality time series to suppress overfitting of nonlinear models. Second, it embeds series decomposition inside the encoder and decoder, which obtains more predictable components from complex time series for long-term prediction. Third, SG-Autoformer utilizes the auto-correlation mechanism to capture the time dependence and improve information utilization. Extensive experiments with real-world datasets show that SG-Autoformer outperforms other advanced prediction methods in terms of prediction accuracy.

In this article, a novel real-time acceleration-continuous path-constrained trajectory planning algorithm is proposed with an appealing built-in tradeoff mechanism between the cruise motion and time-optimal motion. Different from existing approaches, the proposed approach smoothens time-optimal trajectories with bang-bang input structures to generate acceleration-continuous trajectories while preserving the completeness property. More importantly, a novel built-in tradeoff mechanism is proposed and embedded into the trajectory planning framework so that the proportion of the cruise motion and time-optimal motion can be flexibly adjusted by changing a user-specified functional parameter. Thus, the user can easily apply the trajectory planning algorithm for various tasks with different requirements on motion efficiency and cruise proportion. Moreover, it is shown that feasible trajectories are computed more quickly than optimal trajectories. Rigorous mathematical analysis and proofs are presented for those aforementioned theoretical results. Comparative simulations and experimental results on an omnidirectional wheeled mobile robot demonstrate that flexible tunings between the cruise and time-optimal motions can be achieved in a higher computational efficiency manner by the proposed algorithm. Note to Practitioners-This article is motivated by the time-optimal and smooth motion planning problem for mobile robots along given paths. Existing approaches generally use the piecewise polynomial interpolations to smoothen and adjust feasible trajectories. This article proposes a novel path-constrained trajectory planning approach, which preserves properties of completeness and a high-efficient tradeoff mechanism between the optimal and cruise motions when achieving a globally optimal and acceleration-continuous trajectory. Comparative experimental results with other methods show the effectiveness of the proposed approach. In future research, we will attempt to integrate the proposed approach with typical path planning methods to achieve a complete and high-efficient motion planning framework.

Trajectory planning approaches including off-line and on-line algorithms are developed to deal with physical constraints in practical systems. However, by now the existing trajectory planning algorithms have to assume that the desired trajectory function to be planned is fully or at least partly known in advance, and it may not be true in some applications. To overcome this limitation, a general framework of on-line planning a desired trajectory under physical constraints whose function is unknown in advance is proposed. The desired trajectory is first on-line interpolated to achieve a mathematical expression, and then planned under the physical constraints by the bound estimator and nonlinear filter. A heuristic critical test curve algorithm is proposed to solve the potential stability issue. A telerobotic system, where the function of slave-desired trajectory is unknown in advance, is selected as a typical case. The experimental results validate the effectiveness of the proposed planning algorithm.