Journal of Mechanisms Transmissions and Automation in Design · 1985 · 150 citations · 0 references
Robot KinematicsMotion ControlRobot ControlRobotic SystemsJacobian MatrixSingularity FieldEngineeringAerospace EngineeringMechatronicsMechanical SystemsRate DecompositionReference FrameKinematicsMotion ModelingRobotics
In robotics, inverting the Jacobian matrix is essential for rate decomposition and solving inverse kinematics, enabling coordinated motion relative to a fixed reference frame. The paper presents a general methodology for formulating and manipulating the Jacobian matrix, and applies it to analyze geometrically singular positions. The methodology ties the Jacobian formulation to system geometry and simplifies it via suitable coordinate transformations. The approach yields efficient, analytical inverses and enables deduction of general results about the singularity field structure.
Inversion of the Jacobian matrix is the critical step in rate decomposition which is used to solve the so-called “inverse kinematics” problem of robotics. This is the problem of achieving a coordinated motion relative to the fixed reference frame. In this paper a general methodology is presented for formulation and manipulation of the Jacobian matrix. The formulation is closely tied to the geometry of the system and lends itself to simplification using appropriate coordinate transformations. This is of great importance since it gives a systematic approach to the derivation of efficient, analytical inverses. The method is also applied to the examination of geometrically singular positions. Several important general results relating to the structure of the singularity field are deducible from the structure of the algebraic system.