Numerical determination of optimal non-holonomic paths in the presence of obstacles

Abstract

This paper addresses the problem of numerically finding an optimal path for a robot with non-holonomic constraints. A car like robot, whose turning radius is lower bounded i s considered as an example, where the arc length and the change in steering angle are optimized. The car like robot is kinematically constrained and is modelled as a 2 D object translating and rotating in the horizontal plane in the midst of well defined static obstacles. Given the initial and final configurations of the car and a complete description of the obstacles, our procedure directly generates a nonholonomic path as a function of the control variables in an environment of reasonable obstacle clutter. Nonholonomic paths in the midst of more complet obstacle clutter have been generated by identifying grid points on a geometric road map and by applying our procedam between successive grid points.

A method of progressive constraints for nonholonomic motion planning

“…One has to find a control function u(1) that minimizes some functional accounting for the admissibility and the optimality of the corresponding path. This can be transformed into a nonlinear optimization problem and solved with standard variational techniques [7], [20].…”

Real-time collision-free motion planning of a mobile robot using a neural dynamics-based approach

“…The location of the th neuron at the grid in , denoted by a vector , uniquely represents a configuration in , or a location in . The dynamics of each neuron in the network is characterized by a shunting equation derived from (7). Each neuron has a local lateral connections to its neighboring neurons that constitute a subset in .…”

“…The neural activity in the shunting equation is bounded in the finite interval . In the shunting model in (7) or (12), the neural activity increases at a rate of , which is not only proportional to the excitatory input , but also proportional to an auto gain control term . Thus, with an equal amount of input , the closer value of and , the slower increases.…”

Steering a Mobile Robot: Selection of a Velocity Profile Satisfying Dynamical Constraints

“…• When the curvature radius ρ(s) is "large", then the distance along the path between two successive points is equal to a given value ∆L, leading to the following value for ∆s: ∆s = ∆L (X′ 2 (s) + Y′ 2 (s)) (18) In order to ensure smooth transition between these two situations, the threshold value ρ 0 between "small" and "large" values of ρ is choosen as ρ 0 = ∆L α . This implies that the maximum distance between 2 successive points is equal to ∆L.…”

“…The problem has then to be solved globally, as an optimization problem with constraints. To our knowledge, there are only partial solutions: either the analysis is based on the kinematic model, prohibiting therefore to include dynamic constraints (see, for instance, [18]), or there is no geometric constraint on the path (see, for instance [24,22]).…”