Local Geometric Control of a Certain Mechanism with the Growth Vector (4,7)

Abstract

We study local control of the mechanism with the growth vector (4, 7). We study controllability and extremal trajectories on the nilpotent approximation as an example of the control theory on Lie group. We give solutions of the system an show examples of local extremal trajectories.2010 Mathematics Subject Classification. 53C17, 93C15, 34H05.

Note on geometric algebras and control problems with SO(3)-symmetries

“…We focus on two control problems such that their symmetry groups contain SO(3) as subgroups. The first system has the growth vector (3,6) and the other one has the growth vector (4,7), [14,9].…”

“…In particular, possible motions of the mechanism induce a specific filtration in the configuration space. We present two examples that carry the filtration (3,6) and (4,7), respectively, [10,9].…”

“…In nilpotent Figure 1. Generalized trident snakes approximations of our control problems, there is a subgroup of symmetries which is isomorphic to the Lie group SO(3), [11,9]. This leads us to the idea of local control in geometric algebra approach.…”

On symmetries of a sub--Riemannian structure with growth vector $(4,7)$

“…This paper is motivated by the paper [8] where the first and third authors study local control of a planar mechanism with seven-dimensional configuration space whose movement is restricted by three non-holonomic conditions. The mechanism that the authors deal with is a modification of a planar mechanism generally known as trident snake robot.…”

“…Trident snake robot consists of a root block in the shape of an equilateral triangle together with three one-link branches each of which is connected to one vertex of the root block via revolute joint and has a passive wheel at its very end, [9,6]. The generalized trident snake robot introduced in [8] is a planar mechanism consisting of a root block in the shape of an equilateral triangle together with three one-link branches that have passive wheels at their ends, too. However, each of the branches is connected to one vertex of the root block via prismatic joint and one of the joints is simultaneously revolute joint, see Figure 1.…”