Line-integral estimates and motion planning using the continuation method. (English) Zbl 0946.70003

Chitour, Yacine; Sussmann, Héctor J.

Baillieul, John (ed.) et al., Essays on mathematical robots. Proceedings of a workshop, Minneapolis, MN, USA, January 25-29, 1993. New York, NY: Springer. IMA Vol. Math. Appl. 104, 91-125 (1998).

For smooth driftless control-affine systems, the approach to the nonholonomic path-planning problem based on the continuation method leads to an ordinary differential equation in the space of open-loop controls, known as the “path-lifting differential equation” (PLE). Global existence of the solutions to the PLE is known for systems that satisfy Strichartz’s strong bracket generating conditions (SBGC). In the paper this existence result is extended by proving an estimate for certain line integrals along trajectories, and showing that this estimate implies the global existence result not only in the case when the SBGC is satisfied, but also in some other cases. In particular, the proposed approach is applied to a model of front-wheel-driven car (for which the SBGC does not hold), and it is shown that the continuation method can solve the path-planning problem for this model.
For the entire collection see [Zbl 0903.00092].

Reviewer: Bojidar Cheshankov (Sofia)

MSC:

70B15	Kinematics of mechanisms and robots
70B10	Kinematics of a rigid body
70Q05	Control of mechanical systems

Keywords:

abnormal extremals; sub-Riemannian manifolds; path-lifting differential equation; nonholonomic path-planning problem; continuation method; open-loop controls; Strichartz’s strong bracket generating conditions; global existence; front-wheel-driven car