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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.