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Extended Jacobian inverse kinematics algorithm for nonholonomic mobile robots. (English) Zbl 1114.70006

In this paper, the term “nonholonomic mobile robot” denotes a mobile robot whose motion is subject to nonintegrable velocity constraints. The paper restricts to the constraints linear in velocity, resulting in a representation of the mobile robot kinematics in the form of a driftless control system whose number of inputs is smaller than the dimension of its state space.
A novelty of this paper consists in expanding the extended Jacobian method to nonholonomic mobile robots. The main idea of the extended Jacobian method lies in augmenting the original manipulator’s kinematics in such a way that the resulting extended kinematics is (at least locally) regular, and then in defining an inverse kinematics algorithm on the basis of the extended Jacobian. As a point of departure, the endogenous configuration space approach is assumed. Given the mobile robot kinematics and its Jacobian, the authors begin with a decomposition of the endogenous configuration space into a finite-dimensional subspace, isomorphic to the taskspace and the remaining inifinite-dimensional subspace. Next, they augment the original kinematics in order to obtain the extended kinematics which maps the endogenous configuration space into itself. The derivative of the original and augmenting kinematics defines the extended Jacobian that in turn provides a right inverse of the original Jacobian. This inverse is substituted into the Wazewski equation of the continuation method resulting in the extended Jacobian inverse kinematics algorithm.
Besides solving the inverse kinematic problem, the extended Jacobian inverse kinematics algorithms have two additional advantages: they are repeatable, and they offer a possibility of a coordinated control of robot by prescribing a desirable relationship between certain components of control functions. As an illustration of the theory, the authors define a specific, simple extended Jacobian inverse kinematics algorithm and examine its performance by the computer simulations.

MSC:

70B15 Kinematics of mechanisms and robots
70E60 Robot dynamics and control of rigid bodies
70F25 Nonholonomic systems related to the dynamics of a system of particles
70Q05 Control of mechanical systems
93C85 Automated systems (robots, etc.) in control theory

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