Trajectory-based optimal linearization for nonlinear autonomous vector fields

This paper deals with an optimal approximation in the least square sense of nonlinear vector fields. The optimal approximation consists of a linearization along a trajectory that approximates the nonlinear solution from the initial state to the equilibrium position. It is shown that the optimal linearization can be seen as a generalization of the classical linearization. Furthermore, the optimal linearization can approximate the derivative at the equilibrium point, and the order of the method is the same as the nonlinearity, since the approximation depends on the initial state. We also show that the method can be used to study the asymptotic stability of the equilibrium of a nonlinear vector fields, especially in the nonhyperbolic case. Simulation shows good agreement between the linearized and the nonlinear systems.     

Line of sight robot navigation toward a moving goal.

In this paper, we consider the problem of robot tracking and navigation toward a moving goal. The goal's maneuvers are not a priori known to the robot. Thus, off-line strategies are not effective. To model the robot and the goal, we use geometric rules combined with kinematics equations expressed in a polar representation. The intent of the strategy is to keep the robot between a reference point, called the observer, and the goal. We prove under certain assumptions that the robot navigating using this strategy reaches the moving goal successfully. In the presence of obstacles, the method is combined with an obstacle avoidance algorithm. The robot then moves in two modes, the navigation mode and the obstacle avoidance mode. Simulation of various scenarios highlights the efficiency of the method and provides an instructive comparison between the paths obtained for different reference points.     

Successive linearizations of second order multidimensional time-invariant systems

The present paper discusses a linearization method for second order multidimensional time-invariant systems. The method approximates locally the nonlinear vector field around the equilibrium, where the solution starting from a given initial state near the equilibrium is approximated by a linear solution. This linearization is usually called local trajectory-based linearization. The approximation is computed using an iterative method, which consists of successive approximations in the least square sense. Using a numerical example, it is shown that the linearized solutions exhibit good agreement with the nonlinear solutions.     

Modeling and controlling a robotic convoy using guidance laws strategies.

This paper deals with the problem of modeling and controlling a robotic convoy. Guidance laws techniques are used to provide a mathematical formulation of the problem. The guidance laws used for this purpose are the velocity pursuit, the deviated pursuit, and the proportional navigation. The velocity pursuit equations model the robot's path under various sensors based control laws. A systematic study of the tracking problem based on this technique is undertaken. These guidance laws are applied to derive decentralized control laws for the angular and linear velocities. For the angular velocity, the control law is directly derived from the guidance laws after considering the relative kinematics equations between successive robots. The second control law maintains the distance between successive robots constant by controlling the linear velocity. This control law is derived by considering the kinematics equations between successive robots under the considered guidance law. Properties of the method are discussed and proven. Simulation results confirm the validity of our approach, as well as the validity of the properties of the method. Index Terms-Guidance laws, relative kinematics equations, robotic convoy, tracking.