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IV Robotics Miomir Vukobratovi´ c 451 © 2002 by CRC Press LLC 19 Robot Kinematics 19.1 19.2 19.3 19.4 19.5 19.6 19.7 Bruno Siciliano Università degli Studi di Napoli Federico II Introduction Description of Orientation Rotation Matrix • Unit Quaternion • Euler Angles Direct Kinematics Homogeneous Transformation • Denavit-Hartenberg Convention • Joint Space and Task Space Inverse Kinematics Closed-Form Solutions Differential Kinematics Geometric Jacobian • Analytical Jacobian • Singularities Differential Kinematics Inversion Pseudoinverse • Redundancy • Damped Least-Squares Inverse • User-Defined Accuracy Inverse Kinematics Algorithms Jacobian Pseudoinverse • Jacobian Transpose • Use of Redundancy • Orientation Errors 19.8 Further Reading 19.1 Introduction From a mechanical viewpoint, a robotic system generally consists of a locomotion apparatus (legs, wheels) to move in the environment and a. | IV Robotics Miomir VukobratoviC 2002 by CRC Press LLC 19 Robot Kinematics 19.1 Introduction 19.2 Description of Orientation Rotation Matrix Unit Quaternion Euler Angles 19.3 Direct Kinematics Homogeneous Transformation Denavit-Hartenberg Convention Joint Space and Task Space 19.4 Inverse Kinematics Closed-Form Solutions 19.5 Differential Kinematics Geometric Jacobian Analytical Jacobian Singularities 19.6 Differential Kinematics Inversion Pseudoinverse Redundancy Damped Least-Squares Inverse User-Defined Accuracy 19.7 Inverse Kinematics Algorithms Bruno Siciliano Jacobian Pseudoinverse Jacobian Transpose Use of Universitä degli Studi di Napoli Redundancy Orientation Errors Federico II 19.8 Further Reading 19.1 Introduction From a mechanical viewpoint a robotic system generally consists of a locomotion apparatus legs wheels to move in the environment and a manipulation apparatus to operate on the objects present. It is then important to distinguish between mobile robots and robot manipulators. The mechanical structure of a robot manipulator consists of a sequence of links connected by means of joints. Links and joints are usually made as rigid as possible to achieve high precision in robot positioning. The presence of elasticity at the joint transmissions or the use of lightweight materials for the links poses a number of interesting issues that lead to separating the study of flexible robot manipulators from that of rigid robot manipulators. The latter are implicitly meant by the term robots throughout this chapter. This chapter surveys the fundamentals of robot kinematics. Basic mathematical tools such as the rotation matrix the unit quaternion and the Euler angles are briefly recalled. They serve to describe the orientation of the robot s end effector that together with the position can be expressed as a function of the joint variables. This is the direct kinematics equation that is derived through a systematic procedure based on the use of homogeneous .