Jair Koiller
Cidade Universitaria, Juiz de Fora, Brazil, 36036
Departamento de Matemática da UFJF
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 indepth study of the theoretical basis and fundamentals of this field of science.
Publications:
Regis Rodrigues A., Castilho C., Koiller J.
On the Linear Stability of a Vortex Pair Equilibrium on a Riemann Surface of Genus Zero
2022, vol. 27, no. 5, pp. 493524
Abstract
We present a simple procedure to perform the linear stability analysis of a vortex pair equilibrium
on a genus zero surface with an arbitrary metric. It
consists of transferring the calculations to the round sphere in $\mathbb{R}^3$, with a conformal factor,
and exploring the Möbius invariance of the conformal structure, so that the equilibria, seen on the representing sphere, appear in the north/south poles. Three example problems are analyzed: $i)$ For a surface of revolution of genus zero, a vortex pair located at the poles is nonlinearly stable due to integrability.
We compute the two frequencies of the linearization. One is for the reduced system, the other is related to the
reconstruction. Exceptionally, one of them can vanish. The calculation requires only the local profile at the poles and one piece of global information (given by a quadrature). $ii)$ A vortex pair on a double faced elliptical region, limiting case of the triaxial ellipsoid when the smaller axis goes to zero. We compute the frequencies of the pair placed at the centers of the faces. $iii)$ The stability, to a restricted set
of perturbations, of a vortex equilateral triangle located in the equatorial plane of a spheroid, with polar vortices added so that the total vorticity vanishes.

Koiller J., Castilho C., Regis Rodrigues A.
Vortex Pairs on the Triaxial Ellipsoid: Axis Equilibria Stability
2019, vol. 24, no. 1, pp. 6179
Abstract
We consider a pair of opposite vortices moving on the surface of the triaxial ellipsoid
$\mathbb{E}(a,b,c):$ $x^2/a+y^2/b+z^2/c=1, \, a < b < c$. The equations of motion are transported to $S^2 \times S^2$ via a conformal map that combines confocal quadric coordinates for the ellipsoid and spheroconical coordinates in the sphere. The antipodal pairs form an invariant submanifold for the dynamics. We characterize the linear stability of the equilibrium pairs at the three axis endpoints.

Castro A. L., Koiller J.
On the Dynamic Markov–Dubins Problem: from Path Planning in Robotic and Biolocomotion to Computational Anatomy
2013, vol. 18, nos. 12, pp. 120
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 (dMD): to find the timeoptimal $C^2$ 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 dMD 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.

Ehlers K. M., Koiller J.
Microswimming Without Flagella: Propulsion by Internal Structures
2011, vol. 16, no. 6, pp. 623652
Abstract
Since a first proofofconcept for an autonomous microswimming device appeared in 2005 a strong interest on the subject ensued. The most common configuration consists of a cell driven by an external propeller, bioinspired by bacteria such as E.coli. It is natural to investigate whether microrobots 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 rodshaped 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.

Koiller J., Ehlers K. M.
Rubber Rolling over a Sphere
2007, vol. 12, no. 2, pp. 127152
Abstract
"Rubber" coated bodies rolling over a surface satisfy a notwist 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 235 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 235 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 [48] 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 nonclosed 2form $\omega_{NH}$. In this paper we present some basic results on the spheresphere 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 spheroconical 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 spheroconical coordinates separates the Hamiltonian in this case. No other integrable cases with different $I_j$ are known.
