Mobile robotic systems are used in a wide range of applications. Especially in the assistive field, they can enhance the mobility of the elderly and disable people. Modern robotic technologies have been implemented in wheelchairs to give them intelligence. Thus, by equipping wheelchairs with intelligent algorithms, controllers, and sensors, it is possible to share the wheelchair control between the user and the autonomous system. The present research proposes a methodology for intelligent wheelchairs based on head movements and vector fields. In this work, the user indicates where to go, and the system performs obstacle avoidance and planning. The focus is developing an assistive technology for people with quadriplegia that presents partial movements, such as the shoulder and neck musculature. The developed system uses shared control of velocity. It employs a depth camera to recognize obstacles in the environment and an inertial measurement unit (IMU) sensor to recognize the desired movement pattern measuring the user’s head inclination. The proposed methodology computes a repulsive vector field and works to increase maneuverability and safety. Thus, global localization and mapping are unnecessary. The results were evaluated by simulated models and practical tests using a Pioneer-P3DX differential robot to show the system’s applicability.

Let ${\it\gamma}$ be a hyperbolic closed orbit of a $C^{1}$ vector field $X$ on a compact $C^{\infty }$ manifold $M$ and let $H_{X}({\it\gamma})$ be the homoclinic class of $X$ containing ${\it\gamma}$. In this paper, we prove that if a $C^{1}$-persistently expansive homoclinic class $H_{X}({\it\gamma})$ has the shadowing property, then $H_{X}({\it\gamma})$ is hyperbolic.

The article discusses the local solvability (or lack thereof) of various classes of smooth, complex vector fields that vanish on some non-empty subset of the base manifold.

The energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k+1 which is not attained by any non-singular vector field for k>1. For k=1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

We show that the vanishing order of a non-zero vector field at a generic point of a smooth Fano variety of Picard number 1 cannot exceed the dimension of the Fano variety. Furthermore, if there exist only finitely many rational curves of minimal degree through a generic point of the Fano variety, we show that a non-zero vector field cannot vanish at a generic point of the Fano variety.

In this paper we solve the problem of desingularization of an absolutely isolated singularity of a differential equation, including the dicritical case. As an application, we prove the finiteness of the number of dicritical points in the blowing up tree of an absolutely isolated singularity.