球坐标下三维非完整车辆的全局限定指数镇定
Global Exponential Stabilization of a 3D Nonholonomic Vehicle in Spherical Coordinates
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中文总结 AI 辅助
针对受布罗克特条件限制的三维非完整车辆，提出基于球坐标奇异性的反步连续时不变反馈律，实现全局限定指数镇定，并构造严格控制李雅普诺夫函数保证指数稳定性。
中文摘要 AI 辅助
许多空间（三维）车辆，包括自主水下航行器（AUV）和固定翼飞机，实际上是非完整的，并且受到有限驱动能力的限制，因此连续时不变镇定从根本上受到布罗克特必要条件的阻碍。为了克服这一障碍，我们利用球坐标的几何奇异性，设计了一种反步连续、时不变反馈律，该反馈律能够指数镇定仅由前进速度、俯仰角速率和偏航角速率驱动的三维非完整车辆。由此产生的闭环吸引域仅排除了坐标奇异性、余维数为二、测度为零的初始条件集合，在该集合中车辆位于通过目标且垂直于目标平面的直线上，从而覆盖了尽可能大的域。我们进一步构造了一个严格的控制李雅普诺夫函数，以证明在该域上原点具有全局指数稳定性，并具有用户指定的衰减率，同时防止系统接近奇异集。最后，我们证明了闭环系统在笛卡尔坐标下对原点具有指数吸引性。在球坐标和笛卡尔坐标下的数值仿真示例说明了该控制律的有效性。
英文摘要
Many spatial (3D) vehicles, including AUVs and fixed-wing aircraft, are effectively nonholonomic and subject to limited actuation, such that continuous time-invariant stabilization is fundamentally obstructed by Brockett's necessary condition. To overcome this obstruction, we exploit the geometric singularity of spherical coordinates to design a backstepping continuous, time-invariant feedback law that exponentially stabilizes a 3D nonholonomic vehicle actuated solely by forward surge velocity, pitch rate, and yaw rate. The resulting closed-loop region of attraction excludes only the coordinate singularity, codimension-two, measure-zero set of initial conditions in which the vehicle lies on the line through the target orthogonal to the target plane, thereby covering the largest possible domain. We further construct a strict control Lyapunov function to prove global exponential stability of the origin on this domain with a user-specified decay rate, while simultaneously preventing the system from approaching the singular set. Finally, we show that the closed-loop system is exponentially attractive to the origin in Cartesian coordinates. Numerical simulation examples in both spherical and Cartesian coordinates illustrate the effectiveness of the control law.
发表机构
- UC San Diego（加州大学圣地亚哥分校）
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