A definition of the extended Jacobian inverse kinematics algorithm for mobile robots

Using control theoretic concepts we present a definitional procedure of extended Jacobian inverse kinematics algorithms for mobile robots. As a point of departure we assume a representation of the mobile robot kinematics as the end point map of a driftless control system with outputs. The space of admissible control functions of this system plays the role of the configuration space of the mobile robot, the system output space corresponds to the taskspace, so that the mobile robot kinematics transform the configuration space into the taskspace. The extended Jacobian inverse kinematics algorithm is obtained by means of the continuation method, and is based on the extended Jacobian inverse. The main step of its derivation consists in extending the original mobile robot kinematics to a map of the configuration space into itself. To this aim the configuration space is decomposed into a finite dimensional subspace, isomorphic to the taskspace, and the remaining quotient subspace. In compliance with this decomposition an augmenting kinematics map is introduced. The original kinematics and the augmenting kinematics constitute the extended kinematics. In a region free from singularities the inverse of the derivative of the extended kinematics defines the extended Jacobian inverse. By design, the extended Jacobian inverse kinematics algorithm has the property of repeatability. In the paper, the general procedure is exemplified by a derivation of the extended Jacobian inverse for a chained form system that is feedback equivalent to the kinematics of the kinematic car type mobile robot. An examination of algorithmic singularities of this algorithm is carried out. Computer simulations illustrate the performance of the algorithm.