midpoint linearization

This paper addresses the problem of locally linearizing nonlinear systems within experimentally constrained initial-state regions, moving beyond conventional linear assumptions and reliance on a single long trajectory. We propose a finite-sample identification framework that integrates multi-trajectory deterministic sampling with regularized least squares, and establish—for the first time—an explicit error bound quantifying the trade-off between nonlinear approximation error and measurement noise. Theoretically, we prove that the estimated linearization model converges consistently under finite data. Numerical experiments demonstrate that classical i.i.d. single-trajectory excitation methods suffer significant failure risks in nonlinear settings, whereas our approach remains robust and effective. Key contributions are: (1) a localized linearization modeling paradigm tailored to nonlinear systems; (2) a synergistic design of multi-trajectory deterministic sampling and regularized estimation; and (3) the first finite-sample theoretical analysis jointly accounting for both nonlinear model mismatch and statistical estimation error.

This work addresses the challenge of achieving safe, interpretable, and real-time trajectory tracking for domestic service robots while preserving the geometric structure of variables such as SE(3) poses and SPD(n) stiffness/damping matrices—a balance that existing methods struggle to maintain between stability and accuracy. To this end, we propose the Curve-Induced Dynamical System on Manifolds (CDSM), which, for the first time, integrates a curve-induced mechanism into dynamical system modeling on Riemannian manifolds and Lie groups. By decomposing motion into tangential progression and normal attraction components, CDSM unifies stable convergence, online adaptability, and high-precision trajectory generation. Experiments demonstrate that CDSM significantly improves trajectory accuracy, reduces path deviation, and accelerates query speed on the S2 benchmark, with successful real-time adaptive control of both SE(3) and SPD(n) variables validated on robotic arms and mobile platforms.

Modeling and control of high-dimensional nonlinear systems face challenges including substantial model reduction errors and geometric structure mismatches. This paper proposes a fiber-aligned optimal projection method tailored for spectral submanifolds (SSMs), which rigorously derives a non-orthogonal optimal projection satisfying differential-geometric constraints—overcoming the structural limitations of conventional orthogonal projections on nonlinear manifolds—and enabling embedding of controllable systems. The method integrates data-driven SSM learning, geometry-constrained optimization, nonlinear dimensionality reduction, and model predictive control. Evaluated on a 180-dimensional robotic system, it achieves up to a fivefold improvement in trajectory tracking accuracy over state-of-the-art methods, while significantly enhancing long-term prediction fidelity and closed-loop control performance.

Real-time control of high-dimensional nonlinear legged robots (e.g., humanoid and quadrupedal platforms) remains challenging; existing Koopman-based data-driven linearization methods suffer from approximation error, domain shift, and fixed latent-space dimensionality, limiting generalizability and scalability. Method: We propose a continual-learning-driven Koopman dynamical modeling framework that progressively refines linear approximation of true system dynamics via online dataset expansion and adaptive latent-space dimension augmentation. Contribution/Results: We provide the first theoretical proof of monotonic convergence of the linear approximation error. Furthermore, we design the first Koopman-MPC architecture enabling stable locomotion across diverse terrains. Extensive validation on Unitree G1/H1/A1/Go2 and ANYmal D demonstrates that simple linear MPC suffices for robust gait control—significantly enhancing the engineering practicality and cross-terrain generalization capability of Koopman-based control.

To address the challenge of optimal control for nonlinear stochastic systems, this paper proposes the Spectral Dynamics Embedding Control (SDEC) algorithm. SDEC is the first method to deeply integrate finite-dimensional spectral dynamical embedding—grounded in Koopman operator theory—with stochastic optimal control and policy gradient methods, enabling linear representation of the state-value function and effective policy optimization. Its key contributions are: (i) rigorous quantification of both truncation error from finite-dimensional spectral approximation and statistical error from finite-sample estimation; and (ii) incorporation of nonlinear dynamics priors and spectral embedding structure to ensure policy convergence and provable error bounds. Evaluated on the cart-pole swing-up task, SDEC significantly outperforms both Koopman-based linearization and iterative Linear-Quadratic Regulator (iLQR), achieving superior performance while providing quantifiable error bounds and theoretically guaranteed convergence in policy evaluation and optimization.

This work addresses the limitations of classical trajectory planning methods, which prioritize kinematic smoothness while neglecting dynamics and actuator control effort, often resulting in large tracking errors and high energy consumption. To overcome these issues, the authors propose a control-aware optimal trajectory planning framework that explicitly integrates the nonlinear dynamics of robotic manipulators and actuator effort over a finite time horizon. A midpoint linearization strategy is introduced to enhance the accuracy of dynamic approximations during large-range motions. By establishing a unified nonlinear closed-loop simulation environment, the study enables, for the first time, an isolated evaluation of trajectory generation methods under identical conditions. Experimental results on a simplified UR5 model demonstrate that the proposed approach significantly reduces tracking error, corrective torque, and overall closed-loop execution cost, achieving substantially lower energy consumption and total operational expense compared to conventional planners such as cubic, quintic, and trapezoidal profiles.

Nonlinear dynamical systems often lack robust invertibility with respect to disturbances and initial state mismatches, limiting their reliability in control and generative modeling. This work proposes a robustly invertible nonlinear dynamical system framework that, for the first time, constructs a causal inverse system by composing strongly input–output monotone dynamic layers with static orthogonal layers. The resulting recurrent neural network ensures both forward and inverse dynamics are contracting and bi-Lipschitz, and yields a nonlinear minimum-phase/all-pass decomposition. A differentiable parameterization is achieved via bi-Lipschitz recurrent equilibrium networks (BiLipRENs), which significantly enhance performance in trajectory optimization, robust control, and complex distribution generation across tasks such as data-driven internal model control, dynamic surrogate loss learning, and signal-space normalizing flows.

研究探讨了优化器在稳定性边缘时的行为，通过扩展'边缘耦合'到重球和Nesterov动量法来分析其固定点及两点轨道的稳定性。

本文提出一种数据驱动框架，通过将相对度条件纳入学习过程来设计和训练反馈线性化控制器，使用神经李导数替代传统组件，并确保闭环稳定性。

This study addresses the computational bottleneck associated with dynamically constrained sampling of state spaces in feedback control and planning. To overcome this challenge, the work reformulates control as a dynamically constrained sampling problem, establishing a mapping framework that bridges controllability, optimal control theory, and generative modeling. Specifically, it integrates flow matching, normalizing flows, and denoising diffusion techniques to guide system evolution toward target states or distributions. The proposed approach enables efficient reachable set sampling and precise trajectory planning while unifying control-theoretic and generative-modeling paradigms. Furthermore, the authors provide an accessible open-source tutorial to facilitate practical adoption by the research community.