Isotropic compliance in the Special Euclidean Group SE(3)

In this paper, the isotropic compliance property is examined in the Special Euclidean Group SE(3). The relation between the wrench and the resulting twist is examined, considering an end-effector of 6 D.O.F. serial manipulators. Two properties are introduced. The first one, called local isotropic compliance, is verified if the force vector is parallel to the tip point displacement vector, and, at the same time, if the torque vector is parallel to the change of orientation vector. The second one, called screw isotropic compliance, is verified if the wrench screw axis is parallel to the twist screw axis. In the latter case, the wrench and twist screw axes are generally not coincident in the Cartesian space. If they are, the contact point could (screw-B isotropic compliance) or could not (screw-A isotropic compliance) belong to the coincident axes. Four cases are analyzed and classified. Active stiffness regulation is considered to achieve isotropic compliance in a generic configuration. Two arrangements are taken into account for the control system, which acts either in parallel or as a series with the passive joint stiffness. The control stiffness matrix is then determined for both the arrangements and for all the four kinds of isotropic compliance. One detailed example of application is presented and the obtained results are verified by using multi-body dynamic simulation.