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.

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.

This paper aims to develop a new kinematic model, the exponential submanifolds (EXPSs) eΩwith Ω a subspace of se(3) (the Lie algebra of the special Euclidean group SE(3)), for mechanism analysis and synthesis. The EXPSs provide perfect models for many global motion types appearing in the past mechanisms, robotics and kinesiology literatures which cannot be modeled as the well known Lie subgroups or their product. We derive in this paper both the sufficient and necessary conditions on Ω such that eΩis an EXPS, and geometric properties of the EXPS for mechanism analysis and synthesis.

Besides a quantity of well-known positive characteristics, parallel robots also obtain a couple of negative properties which limit their efficient use in certain application areas nowadays. Direct kinematic singularities that constrain the internal workspace are one of these inconvenient properties. This paper establishes a unique index, with physical meaning, that indicates the entrance of the Hexa parallel robot, considering its joint limits, to all its direct kinematic singularities. For this, we complement two methods to detect singularities: the power inspired measure method and the Grassmann geometry method. The applied procedure uses screw theory and line geometry tools to represent movements, actions and lines, respectively, in the robot. This procedure is practically verified in Hexa robot and shows that some singularities are overlapped forming large singularities.