A Generalized Practical Method for Analytic Solution of the Constrained Inverse Kinematics Problem of Redundant Manipulators

This article addresses four important issues relating to practical solutions of the inverse kinematics problem of redundant manipulators. First, a generalized recursive method for systematic derivation of analytic expressions for all possible solutions of any redundant manipulator is presented. The method possesses the advantage of identi fying the linear dependence among joint axes and hence allows all singular configurations to be determined. Sec ond, a joint constraint mapping approach for the inte grated consideration of all joint constraints in the solution procedure of the inverse kinematics problem is presented. The result leads to practical real-time procedures. Map ping of the joint position and actuation constraints onto joint rate space is described. Third, simplification of end- effector velocity equations is shown to be possible for most practical manipulator structures by decomposing the end-effector work space into two orthogonal complement sub-work spaces. The decomposed velocity equations have smaller dimensions and, hence, are easier to solve. In manipulator design, structures can be selected for effi cient kinematics manipulation and simplified end-effector velocity equations. To achieve this purpose, type-synthesis design guidelines are given for efficient decomposition or decoupling of the work space. Fourth, two general approaches are described for optimally resolving the kine matics redundancy. The first approach maximizes the end- effector speed in a prescribed direction, while the second approach minimizes a quadratic objective function defined by the user. Examples on work space decomposition and optimal solution of kinematic redundancy are given. In both cases, expressions for the general analytic solution are derived.