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2013
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# Andrei Ardentov

Program Systems Institute of RAS

## Publications:

 Ardentov A. A., Karavaev Y. L., Yefremov K. S. Euler Elasticas for Optimal Control of the Motion of Mobile Wheeled Robots: the Problem of Experimental Realization 2019, vol. 24, no. 3, pp.  312-328 Abstract This paper is concerned with the problem of optimal path planning for a mobile wheeled robot. Euler elasticas, which ensure minimization of control actions, are considered as optimal trajectories. An algorithm for constructing controls that realizes the motion along the trajectory in the form of an Euler elastica is presented. Problems and special features of the application of this algorithm in practice are discussed. In particular, analysis is made of speedup and deceleration along the elastica, and of the influence of the errors made in manufacturing the mobile robot on the precision with which the prescribed trajectory is followed. Special attention is also given to the problem of forming optimal trajectories of motion along Euler elasticas to a preset point at different angles of orientation. Results of experimental investigations are presented. Keywords: mobile wheeled robot, Euler’s elastica, optimal control, experimental investigations Citation: Ardentov A. A., Karavaev Y. L., Yefremov K. S.,  Euler Elasticas for Optimal Control of the Motion of Mobile Wheeled Robots: the Problem of Experimental Realization, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 312-328 DOI:10.1134/S1560354719030055
 Ardentov A. A., Le Donne E., Sachkov Y. L. Sub-Finsler Geodesics on the Cartan Group 2019, vol. 24, no. 1, pp.  36-60 Abstract This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler $\ell_\infty$ norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that any normal extremal is either bang-bang, or singular, or mixed. Consequently, we study mixed extremals. In particular, we prove that every two points can be connected by a piecewise smooth minimizer, and we give a uniform bound on the number of such pieces. Keywords: Sub-Finsler geometry, time-optimal control, geometric control, Cartan group Citation: Ardentov A. A., Le Donne E., Sachkov Y. L.,  Sub-Finsler Geodesics on the Cartan Group, Regular and Chaotic Dynamics, 2019, vol. 24, no. 1, pp. 36-60 DOI:10.1134/S1560354719010027
 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 Abstract 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 DOI:10.1134/S1560354717080020
 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 Abstract 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 DOI:10.1134/S1560354716070066
 Ardentov A. A. Controlling of a Mobile Robot with a Trailer and Its Nilpotent Approximation 2016, vol. 21, no. 7-8, pp.  775-791 Abstract 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 DOI:10.1134/S1560354716070017