3D IN ROBOTICS

The authors describe a novel technique for computing position and orientation of a camera relative to the last joint of a robot manipulator in an eye-on-hand configuration. It takes only about 100+64 Narithmetic operations to compute the hand/eye relationship after the robot finishes the movement, and incurs only additional 64 arithmetic operations for each additional station. The robot makes a series of automatically planned movements with a camera rigidly mounted at the gripper. At the end of each move, it takes a total of 90 ms to grab an image, extract image feature coordinates, and perform camera extrinsic calibration. After the robot finishes all the movements, it takes only a few milliseconds to do the calibration. A series of generic geometric properties or lemmas are presented, leading to the derivation of the final algorithms, which are aimed at simplicity, efficiency, and accuracy while giving ample geometric and algebraic insights. Critical factors influencing the accuracy are analyzed, and procedures for improving accuracy are introduced. Test results of both simulation and real experiments on an IBM Cartesian robot are reported and analysed

In robotics, to deal with coordinate transformation in three-dimensional (3D) Cartesian space, the homogeneous transformation is usually applied. It is defined in the four-dimensional space, and its matrix multiplication performs the simultaneous rotation and translation. The homogeneous transformation, however, is a point transformation. In contrast, a line transformation can also naturally be defined in 3D Cartesian space, in which the transformed element is a line in 3D space instead of a point. In robotic kinematics and dynamics, the velocity and acceleration vectors are often the direct targets of analysis. The line transformation will have advantages over the ordinary point transformation, since the combination of the linear and angular quantities can be represented by lines in 3D space. Since a line in 3D space is determined by four independent parameters, finding an appropriate type of "number representation" which combines two real variables is the first key prerequisite. The dual number is chosen for the line representation, and lemmas and theorems indicating relavent properties of the dual number, dual vector, and dual matrix are proposed. This is followed by the transformation and manipulation for the robotic applications. The presented procedure offers an algorithm which deals with the symbolic analysis for both rotation and translation. In particular, it can effectively be used for direct determination of Jacobian matrices and their derivatives. It is shown that the proposed procedure contributes a simplified approach to the formulation of the robotic kinematics, dynamics, and control system modeling.