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Andrei Ardentov

Program Systems Institute of RAS


Ardentov A. A., Sachkov Y. L.
Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group
2017, vol. 22, no. 8, pp.  909–936
We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations $\mathbb{R}_+$ and a discrete group of reflections $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i. e., the set of minimizers for each terminal point in the Engel group.
Keywords: sub-Riemannian geometry, optimal control, Engel group, Maxwell strata, cut locus, mobile robot, Euler's elasticae
Citation: Ardentov A. A., Sachkov Y. L.,  Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 909–936
Mashtakov A. P., Ardentov A. A., Sachkov Y. L.
Relation Between Euler’s Elasticae and Sub-Riemannian Geodesics on $SE(2)$
2016, vol. 21, no. 7-8, pp.  832-839
In this note we describe a relation between Euler’s elasticae and sub-Riemannian geodesics on $SE(2)$. Analyzing the Hamiltonian system of the Pontryagin maximum principle, we show that these two curves coincide only in the case when they are segments of a straight line.
Keywords: elastica, sub-Riemannian geodesic, group of rototranslations
Citation: Mashtakov A. P., Ardentov A. A., Sachkov Y. L.,  Relation Between Euler’s Elasticae and Sub-Riemannian Geodesics on $SE(2)$, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 832-839
Ardentov A. A.
Controlling of a Mobile Robot with a Trailer and Its Nilpotent Approximation
2016, vol. 21, no. 7-8, pp.  775-791
This work studies a number of approaches to solving the motion planning problem for a mobile robot with a trailer. Different control models of car-like robots are considered from the differential-geometric point of view. The same models can also be used for controlling a mobile robot with a trailer. However, in cases where the position of the trailer is of importance, i.e., when it is moving backward, a more complex approach should be applied. At the end of the article, such an approach, based on recent works in sub-Riemannian geometry, is described. It is applied to the problem of reparking a trailer and implemented in the algorithm for parking a mobile robot with a trailer.
Keywords: mobile robot, trailer, motion planning, sub-Riemannian geometry, nilpotent approximation
Citation: Ardentov A. A.,  Controlling of a Mobile Robot with a Trailer and Its Nilpotent Approximation, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 775-791

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