Manipulator Kinematics and Dynamics On Differentiable Manifolds: Part I Kinematics

Using basic tools of Euclidean space topology and differential geometry, a regular manipulator configuration space comprised of input and output coordinates and conditions that assure existence of both forward and inverse kinematic mappings is shown to be a differentiable manifold, with valuable analytical and computational properties. For effective use of the manifold structure in support of manipulator analysis and control, four categories of manipulator are treated; (1) serial manipulators in which inputs explicitly determine outputs, (2) explicit parallel manipulators in which outputs explicitly determine inputs, (3) implicit manipulators in which explicit input-output relations are not possible, and (4) compound manipulators that require use of mechanism generalized coordinates to characterize input-output relations. Basic results of differential geometry show that differentiable manifolds in each category are naturally partitioned into maximal, disjoint, path connected submanifolds in which the manipulator is singularity free, hence programmable and controllable. Model manipulators in each of the four categories are analyzed to illustrate use of the manifold structure, employing only multivariable calculus and linear algebra. Computational methods for forward and inverse kinematics and construction of ordinary differential equations of manipulator dynamics on differentiable manifolds are presented in part II of the paper, in support of manipulator control.