{'fraction_non_alphanumeric': 0.027918781725888325, 'fraction_numerical': 0.0025380710659898475, 'mean_word_length': 5.1640625, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Steering laws and continuum models for planar formations

We consider a Lie group formulation for the problem of control of formations. Vehicle trajectories are described using the planar Frenet-Serret equations of motion, which capture the evolution of both vehicle position and orientation for unit-speed motion subject to curvature (steering) control. The Lie group structure can be exploited to determine the set of all possible (relative) equilibria for arbitrary G-invariant curvature controls, where G=SE(2) is a symmetry group for the control law. The main result is a convergence result for n vehicles (for finite n), using a Lyapunov function which for n=2, has been previously shown to yield global convergence. A continuum formulation of the basic equations is also presented.