Управление мобильным роботом с прицепом на основе нильпотентной аппроксимации

We consider a two-driving wheel mobile robot with a trailer on a plane. Possible positions are given by q = (x,y,\theta, \varphi) \in M, where (x,y) \in \mathbb{R}^2 is a midpoint of the robot and \theta, \varphi \in S^1 are angles of orientation of the robot and the trailer. The kinematic model reads as 
\dot{x} = u_1 \cos \theta, \quad \dot{y} = u_1 \sin \theta, \quad \dot{\theta} = u_2, \quad \dot{\varphi} = - u_1 \frac{\sin \varphi}{l_t} - u_2 \left(\frac {l_r \cos \varphi}{l_t} + 1\right), \quad \operatorname{(1)}
 where u_1 and u_2 are, respectively, linear and angular velocities of the robot as controls; l_t and l_r are constants defining the geometry of the hooking up system. The motion planning problem is to find controls u_1(t), u_2(t) that steer (1) from a given initial configuration q_0 \in M to a given final one q_1 \in M, i.e., to find a path q(t), s.t. 
 q (0)= q_0=(x_0, y_0, \theta_0, \varphi_0), \qquad q (t_1) = q_1=(x_1, y_1, \theta_1, \varphi_1). \quad \operatorname{(2)}
The method of nipotent approximation is used. The corresponding nipotent problem is a sub-Riemannian (SR) problem on Engel group. We propose an iterative scheme to solve the motion planning problem (1)-(2), where on each iteration the optimal controls for the nilpotent system are applied to (2). The method converges if q_0 and q_1 are close enough. Moreover, the obtained trajectory is close to optimal in terms of \int_0^{t_1} \sqrt{u_1^2 + \alpha^2 u_2^2} \ d t \to \min, with \alpha>0.

Автор: Andrei Ardentov, Alexey Mashtakov
Дата: 27 июня, 2017

Mathematical Control Theory
with a special session in honor of Gianna Stefani
Porquerolles, June 27th-30th 2017

Веб-сайт: http://mct.univ-tln.fr/...