Analysis and control of a Stewart platform as base motion compensators - Part I: Kinematics using moving frames

Kinematics and its control application are presented for a Stewart platform whose base plate is installed on a floor in a moving ship or a vehicle. With a manipulator or a sensitive equipment mounted on the top plate, a Stewart platform is utilized to mitigate the undesirable motion of its base plate by controlling actuated translational joints on six legs. To reveal closed loops, a directed graph is utilized to express the joint connections. Then, kinematics begins by attaching an orthonormal coordinate system to each body at its center of mass and to each joint to define moving coordinate frames. Using the moving frames, each body in the configuration space is represented by an inertial position vector of its center of mass in the three-dimensional vector space ℝ3, and a rotation matrix of the body-attached coordinate axes. The set of differentiable rotation matrices forms a Lie group: the special orthogonal group, SO(3). The connections of body-attached moving frames are mathematically expressed by using frame connection matrices, which belong to another Lie group: the special Euclidean group, SE(3). The employment of SO(3) and SE(3) facilitates effective matrix computations of velocities of body-attached coordinate frames. Loop closure constrains are expressed in matrix form and solved analytically for inverse kinematics. Finally, experimental results of an inverse kinematics control are presented for a scale model of a base-moving Stewart platform. Dynamics and a control application of inverse dynamics are presented in the part II-paper.

Manifolds

We now embark on the full theory, beginning with the concept of a manifold in differential geometry. The meaning of coordinates and coordinate transformations is carefully explained. The metric and its transformation between coordinate frames is discussed. Riemann normal coordinates are described. The concepts of a tangent space and local flatness are discussed and derived. It is shown how to use the metric to calculate distances, areas and volumes, and to describe submanifolds.

Description of Nonlinear Vortical Flows of Incompressible Fluid in Terms of a Quasi-Potential

As it was shown earlier, a wide class of nonlinear 3-dimensional (3D) fluid flows of incompressible viscous fluid can be described by only one scalar function dubbed the quasi-potential. This class of fluid flows is characterized by a three-component velocity field having a two-component vorticity field. Both these fields may, in general, depend on all three spatial variables and time. In this paper, the governing equations for the quasi-potential are derived and simple illustrative examples of 3D flows in the Cartesian coordinates are presented. The generalisation of the developed approach to the fluid flows in the cylindrical and spherical coordinate frames represents a nontrivial problem that has not been solved yet. In this paper, this gap is filled and the concept of a quasi-potential to the cylindrical and spherical coordinate frames is further developed. A few illustrative examples are presented which can be of interest for practical applications.

Lorentzian SRT-Transformation Factors as Solutions of Oscillation-Equations

Shown is the derivation of Lorentz-Einstein k-factor in SRT as an amplitude-term of oscillation-differential equations of second order.This case is shown for classical Lorentz-factor as solution of an equation for undamped oscillation as well as the developed theorem as a second solution for advanced SRT of fourth order with an equation for damped oscillation-states.This advanced term allows a calculation for any velocities by real rest mass.Also accelerated coordinate -frames are discussed.

Learning Sequential Force Interaction Skills

Learning skills from kinesthetic demonstrations is a promising way of minimizing the gap between human manipulation abilities and those of robots. We propose an approach to learn sequential force interaction skills from such demonstrations. The demonstrations are decomposed into a set of movement primitives by inferring the underlying sequential structure of the task. The decomposition is based on a novel probability distribution which we call Directional Normal Distribution. The distribution allows infering the movement primitive’s composition, i.e., its coordinate frames, control variables and target coordinates from the demonstrations. In addition, it permits determining an appropriate number of movement primitives for a task via model selection. After finding the task’s composition, the system learns to sequence the resulting movement primitives in order to be able to reproduce the task on a real robot. We evaluate the approach on three different tasks, unscrewing a light bulb, box stacking and box flipping. All tasks are kinesthetically demonstrated and then reproduced on a Barrett WAM robot.

A Geodesist's Involvement Into Archaeology; A Beginning of Huge Discoveries

<p>Geodesists have a different mindset; major or minor involvement in mapping, navigation, positioning, surveying, gravity, coordinate frames and systems, geographical information systems, photogrammetry, 3D laser scanning, satellite orbit determination, orbital mechanics, interferometry and many other fields all help in the grand builtup of a resilient scientist who can emerge with an explorer attitude towards various facets of life; archaeology being one of them, needless to say. If this hybrid composite of mindset and attitude is combined with disciplined and smart usage of geophysical 3D imaging instrumentations deployed frequently by treasure hunters, geodesists might be in a very unique position to make a big bang in the world; finding archaeological Black Swans that might serve to rewrite certain narratives of ancient history is something geodesists must consider deeply. In this presentation, an approach and a discovery will be presented.</p>

Reflection and transmission approximations for weak contrast orthorhombic media

Subsurface media are in general anisotropic, and this fact should be taken into account for analyzing reflection and transmission (R/T) coefficients. Orthorhombic (ORT) media are commonly regarded as a practical symmetry system to account for polar anisotropy and azimuthal anisotropy. We have focused on the model made up of two welded ORT half-spaces to analyze the R/T coefficients normalized by vertical energy flux. The two half-spaces have azimuthally nonaligned vertical symmetry planes and are parameterized in two local 3D Cartesian coordinate frames, respectively. The vertical coordinate planes in each local frame coincide with the vertical symmetry planes for the corresponding ORT half-spaces. Under the weak contrast assumption for the two half-spaces, this model is taken as the perturbed model in R/T approximations with the perturbation theory. The unperturbed model is also composed of two unperturbed ORT half-spaces with their different vertical symmetry plane orientations inheriting the counterparts for the perturbed half-spaces above and below the interface, respectively. The unperturbed ORT half-spaces above and below the interface have the same model parameters defined in the two local coordinate frames, respectively. With the perturbations respectively evaluated in the local coordinate frames above and below the interface, the azimuth angle that indicates the local frames’ azimuthal difference is decoupled from the model parameter contrasts. Compared with the traditional approximation method with the perturbation theory in a global coordinate frame, the proposed R/T approximations depend on fewer model parameter discontinuities. We also consider the isotropic background medium under the weak anisotropy assumption. Influences of S-wave singularity points are mitigated by introducing pseudowaves for approximations. Numerical tests are implemented to demonstrate the accuracy.