0
2013
Impact Factor

# Jair Koiller

Av N S Gracas, Xerem, Duque Caxias, RJ, Brazil
INMETRO

Theoretical results, including new control strategies of nonlinear dynamics, are increasingly being used by engineers to design and create mobile robots, nonholonomic manipulators and various floating devices. The creation of such maneuverable robots with a wide action spectrum requires an in-depth study of the theoretical basis and fundamentals of this field of science.

## Publications:

 Castro A. L., Koiller J. On the Dynamic Markov–Dubins Problem: from Path Planning in Robotic and Biolocomotion to Computational Anatomy 2013, vol. 18, no. 1-2, pp.  1-20 Abstract Andrei Andreyevich Markov proposed in 1889 the problem (solved by Dubins in 1957) of finding the twice continuously differentiable (arc length parameterized) curve with bounded curvature, of minimum length, connecting two unit vectors at two arbitrary points in the plane. In this note we consider the following variant, which we call the dynamic Markov–Dubins problem (dM-D): to find the time-optimal $C^2$ trajectory connecting two velocity vectors having possibly different norms. The control is given by a force whose norm is bounded. The acceleration may have a tangential component, and corners are allowed, provided the velocity vanishes there. We show that for almost all the two vectors boundary value conditions, the optimization problem has a smooth solution. We suggest some research directions for the dM-D problem on Riemannian manifolds, in particular we would like to know what happens if the underlying geodesic problem is completely integrable. Path planning in robotics and aviation should be the usual applications, and we suggest a pursuit problem in biolocomotion. Finally, we suggest a somewhat unexpected application to "dynamic imaging science". Short time processes (in medicine and biology, in environment sciences, geophysics, even social sciences?) can be thought as tangent vectors. The time needed to connect two processes via a dynamic Markov–Dubins problem provides a notion of distance. Statistical methods could then be employed for classification purposes using a training set. Keywords: geometric mechanics, calculus of variations, Markov–Dubins problem Citation: Castro A. L., Koiller J.,  On the Dynamic Markov–Dubins Problem: from Path Planning in Robotic and Biolocomotion to Computational Anatomy, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 1-20 DOI:10.1134/S1560354713010012
 Ehlers K. M., Koiller J. Micro-swimming Without Flagella: Propulsion by Internal Structures 2011, vol. 16, no. 6, pp.  623-652 Abstract Since a first proof-of-concept for an autonomous micro-swimming device appeared in 2005 a strong interest on the subject ensued. The most common configuration consists of a cell driven by an external propeller, bio-inspired by bacteria such as E.coli. It is natural to investigate whether micro-robots powered by internal mechanisms could be competitive. We compute the translational and rotational velocity of a spheroid that produces a helical wave on its surface, as has been suggested for the rod-shaped cyanobacterium Synechococcus. This organisms swims up to ten body lengths per second without external flagella. For the mathematical analysis we employ the tangent plane approximation method, which is adequate for amplitudes, frequencies and wave lengths considered here. We also present a qualitative discussion about the efficiency of a device driven by an internal rotating structure. Keywords: bio-inspired micro-swimming devices, Stokes flows, efficiency, Synechococcus Citation: Ehlers K. M., Koiller J.,  Micro-swimming Without Flagella: Propulsion by Internal Structures, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 623-652 DOI:10.1134/S1560354711060050
 Koiller J., Ehlers K. M. Rubber Rolling over a Sphere 2007, vol. 12, no. 2, pp.  127-152 Abstract "Rubber" coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied by "marble" coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2-3-5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group $G_2$). The 2-3-5 nonholonomic geometries are classified in a companion paper [2] via Cartan's equivalence method [3]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [4-8] with $SO$(3) symmetry group, total space $Q = SO(3) \times S^2$ and base $S^2$, that can be reduced to an almost Hamiltonian system in $T^*S^2$ with a non-closed 2-form $\omega_{NH}$. In this paper we present some basic results on the sphere-sphere problem: a dynamically asymmetric but balanced sphere of radius $b$ (unequal moments of inertia $I_j$ but with center of gravity at the geometric center), rubber rolling over another sphere of radius $a$. In this example $\omega_{NH}$ is conformally symplectic [9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power $p = 1/2 (b/a - 1)$. Using sphero-conical coordinates we verify the result by Borisov and Mamaev [10] that the system is integrable for $p = -1/2$ (ball over a plane). They have found another integrable case [11] corresponding to $p = -3/2$ (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates separates the Hamiltonian in this case. No other integrable cases with different $I_j$ are known. Keywords: nonholonomic mechanics, reduction, Chaplygin systems Citation: Koiller J., Ehlers K. M.,  Rubber Rolling over a Sphere, Regular and Chaotic Dynamics, 2007, vol. 12, no. 2, pp. 127-152 DOI:10.1134/S1560354707010054