Abstract: -
The control problem of a tentacle manipulator using a robust 3 D visual servoing is presented. Thetheoretical model of this class of arms is studied. Servoing is based on binocular vision obtained from twocameras that ensure a continuous measure of the arm parameters. The control errors function is built in 3Dcartesian space from the visual information obtained in the two image planes. The 2D errors are determined asshape errors, they are calculated as the differences between the actual and desired continuous angle values. Aspatial error is determined and a control law is discussed. Computer simulations and real 3-D experiments arepresented in order to show the applicability of the method.Key-Words: Videoservoing, Robot Control, Hyperredundant Manipulator.
1 Introduction
An ideal tentacle manipulator is a nonconventionalrobotic arm with an infinite mobility. Ithas the capability of takeing sophisticated shapesand of achieving any position and orientation in a3D space. These systems are also known ashyperredundant manipulators and, over the pastseveral years, there has been a rapid expandinginterest in their study and construction.The control of these systems is verycomplicated and a great number of researchers triedto offer solutions for this difficult problem. In [1] itanalyses the control by cables or tendons meant totransmit forces to the elements of the arm in order toclosely approximate the arm as a truly continuousbackbone. Also, Mochiyama has investigated theproblem of controlling the shape of an HDOF rigidlinkrobot with two-degree-of-freedom joints usingspatial curves [7], [8]. Important results wereobtained by Chirikjian and Burdick [3] – [6] wholaid the foundations for the kinematic theory ofhyperredundant robots. Their results are based on a“backbone curve” that captures the robot’smacroscopic geometric features.The inverse kinematic problem is reduced todetermining the time varying backbone curvebehaviour. New methods for determining “optimal”hyper-redundant manipulator configurations basedon a continuous formulation of kinematics aredeveloped. In [2], Gravagne analysed the kinematicmodel of “hyper-redundant” robots, known as“continuum” robots. Robinson and Davies [8]present the “state of art” of continuum robots,outline their areas of application and introduce somecontrol issues. The great number of parameters,theoretically an infinite one, makes very difficult theuse of classical control methods and theconventional transducers for position andorientation.In this paper the method of image-basedservoing [12,15] for a hyperredundant arm isstudied. Servoing is based on binocular vision. Acontinuous measure of the arm parameters, derrivedfrom the real-time computation of the binocularoptical flow over the two images, is compared withthe desired position of the arm.The control error function is built in 3Dcartesian space using the visual informationobtained from two cameras in two image planes[13]. The two 2D errors obtained in the two imageplanes are determined by the two differencesbetween the actual and desired continuous anglevalues that define the projections of the arm shape.The plane errors can be considered as errors of thearm shape. These errors are used to calculate thespatial error and a control law is synthesized.For the closed-loop control system, the stabilityis proven by using the Lyapunov second method.The error function is computed virtually in theimage spaces and the fact that no calibration(camera parameters) is required allows the synthesisof a more robust control laws.
2 Background
Consider a 3D hyperredundant robot with a threedimensional Cartesian coordinate frame called therobot coordinate frame whose axes are labeled X, Y,Z. The mechanical structure represents an ideal arm,with an uniform distributed mass and torque, withideal flexibility that can take any arbitrary shape(Fig. 1). We will neglect friction and structuredamping. The essence of the model is a 3 –dimensional backbone curve C that is parametricallydescribed by a vector r( s )∈ℜ3 and an associateframe Φ( s )∈ℜ3 whose columns create the framebase (Fig. 2) [2,3]. The independent parameter s isrelated to the arc length from the origin of the curveC. The position of a point s on the curve C is definedby the position vector,

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