Frenet oscillations and Frenet–Euler angles: curvature singularity and motion-trajectory analysis
Frenet oscillations and Frenet–Euler angles: curvature singularity and motion-trajectory analysis
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DOI：
10.1007/s11071-021-06798-1
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发表时间：
2021-03
影响因子：
5.6
通讯作者：
A. Shabana
中科院分区：
文献类型：
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作者：
A. Shabana
运动轨迹（MT）曲线被用来引入Frenet振荡。包含绝对速度和加速度矢量的运动平面的时变方向由三个Frenet-Euler角定义：曲率角、垂直展开角和倾斜角，简称为Frenet角。运动平面的Frenet倾斜角和Frenet超高是衡量离心惯性力与水平面的偏差的指标，可以用来解释实际中平衡速度的定义。本文引入了预超高架密切（PSEO）平面的概念，并利用Rodrigues公式导出了一个正交旋转矩阵，它提供了PSEO平面的几何解释。一个新的逆动力学问题，利用实验或模拟记录的运动轨迹（RMT）被用来定义的Frenet惯性力，并证明其等效的笛卡尔形式的惯性力。导出了曲率向量的速度和加速度表达式、给定前进速度时切向加速度的限制、离心力保持水平所需的条件和曲率奇点条件。Frenet倾斜角可以用来证明曲率奇点处法向量的存在性。结果表明，惯性力可以采取不同的形式，这取决于所使用的曲线参数。一个简单的分析曲线的结果演示Frenet振荡和区分的重要性公路坡道和铁路轨道银行的角度和超高，这是时不变的，和Frenet银行的角度和超高，这是运动相关的。
Motion-trajectory(MT) curves are used to introduceFrenet oscillations. Time-varying orientation of themotion planethat contains the absolute velocity and acceleration vectors is defined in terms of threeFrenet–Euler angles; thecurvature,vertical-development, andbank angles, referred to as theFrenet anglesfor brevity. The Frenet bank angle and the associatedFrenet super-elevationof the motion plane, which measure deviation of the centrifugal inertia force from the horizontal plane, can be used to shed light on definition of thebalance speedused in practice. The concept of thepre-super-elevated osculating (PSEO) planeis introduced andRodrigues’ formulais employed to develop an orthogonal rotation matrix that provides a geometric interpretation of the PSEO plane. A newinverse-dynamics problemthat utilizes experimentally or simulationrecorded motion trajectories(RMT) is used to define theFrenet inertia forcesand demonstrate their equivalence to the Cartesian form of the inertia forces. New expressions for the curvature vector in terms of the velocity and acceleration, limit on the magnitude of the tangential acceleration for a given forward velocity, condition required for the centrifugal force to remain horizontal, and condition ofcurvature singular pointsare derived. The Frenet bank angle can be used to prove existence of the normal vectors at the curvature singular points. It is shown that the inertia force can assume different forms, depending on the curve parameter used. The results of a simple analytical curve demonstrateFrenet oscillationsand importance of distinguishing between the highway-ramp and railroadtrack bank anglesand super-elevations, which are time-invariant, and theFrenet bank angleand super-elevation, which are motion-dependent.