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An approach to the planning of optimal robotic motions in the pres ence of obstacles is proposed. It is based on the use of nonclassical formulation of Pontryagin's maximum principle, which makes it possible to handle efficiently the state constraints resulting from the robotic tasks to be performed. The convergence properties of the algorithm are examined. A computer example involving a pla nar redundant manipulator of three revolute kinematic pairs, which performs two tasks in a two-dimensional work space including ob stacles, is given. A comparison of the proposed approach with the well-known method of penalty function is made.

This paper presents the solution at the control feedback level to the inverse kinematics problem for mobile manipulators operating in both obstacle-free task spaces and task spaces including obstacles. Using the Frechet differential of a certain criterion function, the fully specified system of algebraic and differential equations of the minimal amount has been obtained to solve the inverse kinematics problem. Based on the Lyapunov stability theory, a full differential form generating the mobile manipulator trajectory, whose attractor attained in a finite time fulfills the above system of algebraic and differential equations, has been derived. The problem of both singularity and collision avoidance is solved here based on a concept of (local) velocity perturbation which results in continuous mobile manipulator velocities near singularities and obstacles. The numerical simulation results carried out for a mobile manipulator consisting of a nonholonomic wheel and a holonomic manipulator of two revolute kinematic pairs, operating in both an obstacle-free task space and task space including obstacles, illustrate the trajectory performance of the proposed solution scheme.

This paper is concerned with a general learning (with arbitrary criterion and state-dependent constraints) of continuous trajectories by means of recurrent neural networks with time-varying weights. The learning process is transformed into an optimal control framework, where the weights to be found are treated as controls. A new learning algorithm based on a variational formulation of Pontryagin's maximum principle is proposed. This algorithm is shown to converge, under reasonable conditions, to an optimal solution. The neural networks with time-dependent weights make it possible to efficiently find an admissible solution (i.e., initial weights satisfying state constraints) which then serves as an initial guess to carry out a proper minimization of a given criterion. The proposed methodology may be directly applicable to both classification of temporal sequences and to optimal tracking of nonlinear dynamic systems. Numerical examples are also given which demonstrate the efficiency of the approach presented.

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Repetitive flicker stimulation (photic driving) offers the possibility to study the properties and coupling characteristics of stimulation-sensitive neuronal oscillators by means of the MEG/EEG analysis. With flicker frequencies in the region of the individual alpha band frequency, the dynamics of the entrainment process of the alpha oscillation, as well as the dynamics of the accompanying gamma oscillations and the coupling between the oscillations, are investigated by means of an appropriate combination of time-variant analysis methods. The Hilbert and the Gabor transformation reveal time-variant properties (frequency entrainment, phase locking, and n:m synchronization) of the entrainment process in the whole frequency range. Additionally, time-variant partial directed coherence is applied to identify ocular saccadic interferences and to study the directed information transfer between the recording sites of the simultaneously derived MEG/EEG data during the entrainment. The MEG data is the focus of this methodological study as the entrainment effects of the alpha oscillation are stronger in MEG than in the EEG. The occipital brain region (visual cortex) was mainly investigated and the dynamics of the alpha entrainment quantified. It can be shown that at the beginning of this entrainment, a transient, strongly phase-locked "40-Hz" gamma oscillation occurs.