geometric curve optimization

This work addresses the challenge of high-fidelity motion simulation for limbless, deformable organisms—such as snakes—in computer graphics and soft robotics. The authors propose a differential-geometry-based optimization framework that models slender soft bodies as three-dimensional parametric curves. Shape representation is achieved through a Fourier–Chebyshev polynomial basis, while physically plausible and self-intersection-free configurations are enforced via bending and torsion energy constraints. Realistic visual rendering is subsequently obtained by interpolating from the optimized curve to a surface mesh. The method demonstrates strong robustness and generalization in complex environments, significantly outperforming existing approaches in producing more realistic and higher-quality simulations of limbless locomotion.

Nonlinear parameterizations (e.g., Euler angles, rational kinematics) in Graph-Convex Set (GCS) trajectory optimization induce metric distortion in configuration space, degrading trajectory quality and violating geometric fidelity. Method: We propose the first rigorous GCS optimization framework supporting nonconvex objective functions. Our approach introduces a “de-distortion” mechanism that integrates Lagrangian duality with certified collision-free region verification, thereby recovering the true configuration-space metric while preserving original constraint feasibility and theoretical guarantees. Contribution/Results: This work establishes the first tight, verifiable optimization support for nonconvex objectives within the GCS paradigm. Experiments across three canonical robotics scenarios—bimanual manipulation, 3D rotational planning, and rational-kinematic modeling—demonstrate significant reductions in path length and execution time, with only marginal increases in computational overhead.

本文提出一种新方法，通过逆函数定理计算解析IK参数化的梯度，以解决机器人在运动学约束下的轨迹规划问题。

This work addresses gait optimization and motion planning for mobile systems with hybrid kinematic–dynamic characteristics—such as nonholonomic wheeled robots and biomimetic swimming robots. We propose a geometric modeling and optimization framework that unifies second-order dynamics with nonholonomic constraints. For the first time, we integrate Lie group numerical integrators with Lagrangian reduction, leveraging manifold symmetries to enable variational gait optimization. The framework supports anisotropic added inertia and fluid drag modeling, overcoming limitations of Euclidean-space-based optimization. Evaluated on roller racer, snakeboard, and Purcell swimmer platforms, our method efficiently generates diverse locomotion behaviors—including acceleration, steady-state cruising, steering, and smooth multi-gait transitions. Both simulation and physical experiments demonstrate high accuracy and strong generalizability across distinct underactuated, nonholonomic systems.

This work addresses the common omission of fundamental physical constraints—such as inertia, gravity, and viscous drag—in robot motion optimization. We propose a unified optimal control framework grounded in differential geometry. Methodologically, we model viscous drag as a Riemannian metric on the configuration manifold, thereby unifying kinetic energy and gravitational potential fields, and derive geometric optimal control equations incorporating curvature effects. Indirect optimal control is solved via Lagrangian mechanics and manifold-based variational calculus. Experiments on a two-link planar manipulator and a UR5 robot demonstrate that the proposed model substantially alters optimal trajectory shape and energy distribution, enhancing both physical realizability and energy efficiency. Our core contribution is a novel geometric modeling paradigm that synergistically integrates drag, curvature, and potential fields—establishing a new theoretical foundation for physics-informed robotic trajectory optimization.

This work addresses the issue of excessively tortuous trajectories commonly produced by existing feedback motion planners based on simplicial complex decomposition, which lead to slow motion and high energy consumption. The authors propose a novel approach that constructs a safe, goal-directed “funnel” region by heuristically aligning local vector fields and building a goal-centric maximal star-shaped simplicial chain. By integrating systematic vector field alignment with geometric star-chain construction, the method significantly enhances path smoothness and control efficiency while preserving formal safety guarantees. Experimental results demonstrate a 91.40% average reduction in total path curvature and a 45.47% decrease in LQR control energy cost. In low-dimensional configuration spaces, the planner outperforms sampling- and optimization-based alternatives in both computational efficiency and robustness.

This work addresses the problem of smooth path tracing under simultaneous upper and lower curvature bounds by proposing a novel curvature-bounded geodesic model. Formulated within the Hamilton-Jacobi-Bellman (HJB) partial differential equation framework, the model introduces bilateral curvature constraints into the HJB formalism for the first time, enabling strong control over the geometric properties of generated paths. An efficient numerical discretization scheme is devised to balance path smoothness, rigidity, and elasticity. Experimental results demonstrate that the method robustly produces high-quality, curvature-constrained optimal paths in applications such as robotic motion planning and image curve structure tracking, significantly extending the capabilities of conventional single-bound constrained models.

This study addresses the problem of efficiently and robustly approximating arbitrary spatial curves with three-dimensional elastic curves. Leveraging the physical characterization of elastic curves as critical points of the bending energy functional and their equivalence to the spherical pendulum equations, the authors construct an 11-dimensional parametric representation and introduce the first numerically stable inverse solver to robustly recover parameters from a given curve segment. The proposed method significantly enhances both fitting accuracy and computational efficiency, enabling interactive design workflows. It has been successfully applied to rationalizing CAD surfaces for robotic hot-blade cutting, demonstrating high precision, numerical stability, and practical utility in real-world manufacturing scenarios.

This work addresses the challenge of generating real-time, collision-free, and dynamically feasible trajectories for autonomous vehicles in complex environments by proposing a structured optimization approach based on Graphs of Convex Sets (GCS). The method models free space as a GCS and integrates Bézier curve path parameterization with polynomial time scaling within each convex region, embedding trajectory constraints under a simplified bicycle model and linear tire assumptions. By reformulating the nonlinear optimal control problem as a graph-based optimization that preserves convexity through continuous relaxation, the approach effectively unifies geometric and dynamic constraints. Experimental results demonstrate that the generated trajectories achieve efficient static obstacle avoidance and lane changes in CommonRoad scenarios, attaining solution accuracy comparable to nonlinear programming while significantly improving computational efficiency and reducing sensitivity to initial conditions.

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.