Orthogonal Manifold Foliations for Impedance Control of Redundant Kinematic Structures

Abstract

Redundant manipulators have more degrees of freedom then minimally required in order to perform the main manipulation task. This provides more flexibility during the manipulation task as it allows for simultaneous secondary tasks like obstacle avoidance or modification of dynamical properties. However, the overall system becomes underdetermined and methods for redundancy resolution are required. One technique for redundancy resolution is task space augmentation. Besides the task space coordinates, another set of coordinates is defined to be used to determine the configuration of a manipulator. Linear projection is used to ensure that the main task is not disturbed by the new coordinates. In this thesis a novel kind of task space augmentation is designed, which is based on dynamical decoupling. By construction, these new coordinates are dynamically independent and no projection is required. Controllers in both sets of coordinates can be superimposed without mutual interference. The additional new set of coordinates is computed by a coordinate function with certain properties. The mapping from joint space to task space can be seen as a foliation of the joint space manifold, where the leaves correspond to the self-motion manifolds. Based thereon, relations of the Jacobian between the task space forward kinematics and the Jacobian of the desired coordinate function are derived. These relations can be described as an underdetermined system of partial differential equations. In order to find an approximate solution to this, a variational principle is employed. In particular, the desired coordinate function is written as a neural network and the derived requirements on the Jacobians are translated to a cost function. Training of the neural network simultaneously finds a concrete instantiation of the PDE as well as a solution to it. Trained models for different planar robots are evaluated in different settings. Kinematic evaluation shows decoupling of the two sets of coordinates on first-order dynamics, which is generally not provided by traditional augmentation methods. Afterwards, the model is evaluated using simulation of closed-loop dynamics. Impedance controllers in both coordinate sets control a simulated planar robot. In contrast to the kinematic analysis some couplings are observable on actual multi-body dynamics. The majority of couplings is due to Coriolis and centrifugal forces and terms related to the change of the Jacobian. An additional feed-forward controller compensating the major couplings achieves dynamically decoupled coordinates. The developed method provides a technique to automatically find dynamically decoupled coordinates, which can be used for impedance control of redundant robots. These can also be interpreted as providing potentials and geodetic springs which are advantageous for controller design.