Sub-Riemannian problem on the Engel group

The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent (non-linear) approximation to generic nonholonomic systems in four-dimensional space with two dimensional control, for instance to a system which describes motion of a mobile robot with a trailer. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping. The Maxwell strata corresponding to the symmetries are described by certain equations in elliptic functions. We study solvability of these equations, give sharp estimates of their roots, and describe their mutual disposition via the analysis of the elliptic functions involved. The problem of finding optimal synthesis in the general case is reduced to a system of three algebraic equations in elliptic functions and elliptic integrals. It seems impossible to analytically solve such equations, therefore, a software for computing optimal trajectories for the nilpotent sub-Riemannian problem on the Engel group is being developed in Wolfram Mathematica. This software has already been used to devise several algorithms for computing approximate paths close to optimal for a mobile robot with a trailer. Those algorithms will be applied for controlling a real mobile robot with a trailer.

举办单位：数学与统计学院

发 布 人：吴双 发布时间： 2019-05-30

发 布 人：吴双 发布时间： 2019-05-30

Andrei Ardentov received M.S. at University of Pereslavl in 2009 and Ph.D. at Program Systems Institute of Russian Academy of Science (PSI RAS) in 2012 both under supervision of Prof. Sachkov, with whom he is working since 2005. Currently he is a senior researcher of Control Processes Research Centre at PSI RAS, Pereslavl-Zalessky. His primary research goals are directed toward developing geometric control theory and applying it to real-world. He has investigated several nonholonomic control systems on Lie groups and associated sub-Riemannian geometry problems which appear in robotics and computer vision. Since 2006, he is studying classical variational problem about stationary configurations of an elastic rod, which was considered by Leonard Euler in 1744 and by Max Born in 1906 in his Ph.D. thesis, some questions in this problem still remain open.