Harald Löwe

Inversion Symmetry of the Euclidean Group: Theory and Application to Robot Kinematics

Just as the 3-D Euclidean space can be inverted through any of its points, the special Euclidean group SE(3) admits an inversion symmetry through any of its elements and is known to be a symmetric space. In this paper, we show that the symmetric submanifolds of SE(3) can be systematically exploited to study the kinematics of a variety of kinesiological and mechanical systems and, therefore, have many potential applications in robot kinematics. Unlike Lie subgroups of SE(3), symmetric submanifolds inherit distinct geometric properties from inversion symmetry. They can be generated by kinematic chains with symmetric joint twists. The main contribution of this paper is: 1) to give a complete classification of symmetric submanifolds of SE(3); 2) to investigate their geometric properties for robotics applications; and 3) to develop a generic method for synthesizing their kinematic chains.

POE-Based Robot Kinematic Calibration Using Axis Configuration Space and the Adjoint Error Model

The product of exponential model based robot calibration approach eliminates parameter discontinuity and simplifies coordinate frame setup, but demands extra effort to normalize twist coordinates and differentiate parameter-varying exponential maps. In this paper, we show that such an endeavor can be exempted by respecting the nonlinear geometry of the joint axis configuration space (ACS), the set of all possible axis locations. We analyze the geometry of the ACS models for prismatic and revolute joints, and treat the errors as Adjoint transformations on joint twists. We propose a novel robot kinematic calibration algorithm based on the ACS and Adjoint error model. It is geometrically intuitive, computationally efficient, and can easily handle additional assumptions on joint axes relations. We present a comparative study with simulations and experiments to show that our algorithm outperforms the existing ones in various aspects.

Inner ideals and intrinsic subspaces of linear pair geometries

Abstract We introduce the notion of intrinsic subspaces of linear and affine pair geometries, which generalizes the one of projective subspaces of projective spaces. We prove that, when the affine pair geometry is the projective geometry of a Lie algebra introduced in [W. Bertram, K.-H. Neeb, Projective completions of Jordan pairs. I. The generalized projective geometry of a Lie algebra. J. Algebra 277 (2004), 474–519. MR2067615 (2005f:17031) Zbl 02105235], such intrinsic subspaces correspond to inner ideals in the associated Jordan pair, and we investigate the case of intrinsic subspaces defined by the Peirce-decomposition which is related to 5-gradings of the projective Lie algebra. These examples, as well as the examples of general and Lagrangian flag geometries, lead to the conjecture that geometries of intrinsic subspaces tend to be themselves linear pair geometries.

Self-Orthogonal Compact Spreads and Unitals in Topological Translation Planes

AbstractA spread of

Remanufacturing the Pinion: An Application of a New Design Method for Spiral Bevel Gears

V = \mathbb{R}^{2l}

A Design Method for the Pinion of a Spiral Bevel Gear Drive Admitting Full Control on the Contact Area and the Transmission Error

is a set of l-dimensional subspaces L ⩽ V partitioning V ∖ {0}. We construct examples of compact spreads that are identical with their sets of orthogonal spaces L⊥. In the corresponding topological translation planes, every Euclidean sphere is a unital with the additional property that every point at infinity has flat feet.

Applications of the Exponential Function of the Euclidean Motion Group to the Theory of Gearing

Abstract Being a Lie group, the group SE(3) of orientation preserving motions of the real Euclidean 3-space becomes a symmetric space (in the sense of O. Loos) when endowed with the multiplication µ(g, h) = gh−1g. In this note we classify all connected symmetric subspaces of SE(3) up to conjugation. Moreover, we indicate some of its important applications in robot kinematics.

Maximality Properties of Stable Partial Spreads and of Shear Planes

Damages of a large spiral bevel gear drive as used in heavy industry typically affect the pinion. Even if the gear still could be used, the complete pair has to be changed. This leads to long off times, high costs, and unnecessary waste. This paper applies a recent design technology for spiral bevel gears to the production of a replacement pinion for the sake of energy saving, reduction of costs and off times, and for the realization of green engineering. The process involves the following steps. First, the real tool surface of the gear is measured by a CMM. Based on the new design method, the tooth surface of the mating pinion is derived from this discrete point cloud. In order to improve the meshing performance, the resulting surface of the pinion is modified in the third step. Finally, the pinion is produced on a CNC machining center. In contrast to other approaches, none of these steps needs the parameters of the special machine tool defining the original gear pair. It is worth noting that our technology can also be profitable to gain more freedom in the design of new gear pairs.