Existing trajectory planning methods often face challenges in ensuring stable robot motion control, leading to significant positional errors during navigation. This study proposes Timed-Elastic-Band-Based Variable Splitting (TEB-VS), a novel framework that integrates variable splitting (VS)—a constrained optimization technique—with the classical Timed-Elastic-Band (TEB) algorithm. Unlike incremental modifications to TEB, TEB-VS introduces a systematic combination of VS and TEB to decompose non-convex global constraints into tractable subproblems while leveraging symmetry principles for balanced multi-objective control (e.g., velocity, acceleration, and obstacle avoidance). Experimental results demonstrate that TEB-VS achieves a $46.5\%$ improvement in motion stability over traditional TEB in obstacle-free simulations and a $37\%$ enhancement in dynamic obstacle scenarios. Real-world tests show a $26.7\%$ reduction in angular velocity oscillations, with computational efficiency comparable to TEB. The framework’s effectiveness in harmonizing trajectory smoothness and dynamic adaptability is validated through extensive simulations and TurtleBot2 experiments.

Recent advancements in lightweight electroencephalogram(EEG) signal classification have enabled real-time human–robot interaction, yet challenges persist in balancing computational efficiency and safety in dynamic path planning. This study proposes an EEG-based inverse reinforcement learning (EIRL) framework to simulate human navigation strategies by decoding neural decision preferences. The method integrates a pruned WNFG-SSCCNet-ADMM classifier for EEG signal mapping, apprenticeship learning for reward function extraction, and Q-learning for policy optimization. Experimental validation in an 8 × 8 FrozenLake-v1 environment demonstrates that EIRL reduces average path risk values by 50% compared with traditional reinforcement learning, achieving expert-level safety ($\Delta$ = 4) while maintaining optimal path lengths. The framework enhances adaptability in unknown environments by embedding human-like risk aversion into robotic planning, offering a robust solution for applications requiring minimal prior environmental knowledge. Results highlight the synergy between neural feedback and computational models, advancing inclusive human–robot collaboration in safety-critical scenarios.

This review surveys the mathematical theory and methods of state estimation for stochastic dynamical systems when the Gaussian-linear-least-squares paradigm fails, focusing on the role of sparsity-promoting and constraint-encoding regularization. The exposition is organized around a single regularized estimation functional in which the weighted least-squares terms of classical filtering and smoothing are augmented by nonsmooth penalty functions and indicator functions of constraint sets. Bayesian recursion, Gaussian approximation, sequential Monte Carlo, and moving-horizon optimization emerge as different structural assumptions on this common objective. Recent work is reviewed through five relaxations of the classical assumptions: heavy-tailed and non-Gaussian noise, unknown or misspecified models, distributed and networked architectures, unused structural prior information such as sparsity and safety constraints, and switching or unknown external inputs. Emphasis is placed on variable-splitting and augmented-Lagrangian techniques that interface Gaussian recursions with nonsmooth optimization, on convergence and stability guarantees, and on the mathematical challenges that remain open, including convergence rates for alternating minimization, statistical efficiency relative to least-squares baselines, and provable guarantees for learning-enhanced filters.