Involutive moving frames II; The Lie-Tresse theorem

Since opportunities for spaceflight experiments are scarce, ground-based microgravity simulation devices (MSDs) offer accessible and economical alternatives for gravitational biology studies. Among the MSDs, the random positioning machine (RPM) provides simulated microgravity conditions on the ground by randomizing rotating biological samples in two axes to distribute the Earth’s gravity vector in all directions over time. Real-time microscopy and image acquisition during microgravity simulation are of particular interest to enable the study of how basic cell functions, such as division, migration, and proliferation, progress under altered gravity conditions. However, these capabilities have been difficult to implement due to the constantly moving frames of the RPM as well as mechanical noise. Therefore, we developed an image acquisition module that can be mounted on an RPM to capture live images over time while the specimen is in the simulated microgravity (SMG) environment. This module integrates a digital microscope with a magnification range of 20× to 700×, a high-speed data transmission adaptor for the wireless streaming of time-lapse images, and a backlight illuminator to view the sample under brightfield and darkfield modes. With this module, we successfully demonstrated the real-time imaging of human cells cultured on an RPM in brightfield, lasting up to 80 h, and also visualized them in green fluorescent channel. This module was successful in monitoring cell morphology and in quantifying the rate of cell division, cell migration, and wound healing in SMG. It can be easily modified to study the response of other biological specimens to SMG.

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Generalized evolutes of planar curves

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887821502224 ◽

2021 ◽

Author(s):

Yongqiao Wang ◽

Yuan Chang ◽

Haiming Liu

Keyword(s):

Euclidean Plane ◽

Singular Points ◽

Moving Frames ◽

Planar Curves ◽

Fixed Direction ◽

Spatial Curves

The evolutes of regular curves in the Euclidean plane are given by the caustics of regular curves. In this paper, we define the generalized evolutes of planar curves which are spatial curves, and the projection of generalized evolutes along a fixed direction are the evolutes. We also prove that the generalized evolutes are the locus of centers of slant circles of the curvature of planar curves. Moreover, we define the generalized parallels of planar curves and show that the singular points of generalized parallels sweep out the generalized evolute. In general, we cannot define the generalized evolutes at the singular points of planar curves, but we can define the generalized evolutes of fronts by using moving frames along fronts and curvatures of the Legendre immersion. Then we study the behaviors of generalized evolutes at the singular points of fronts. Finally, we give some examples to show the generalized evolutes.

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A Moving Frame Trajectory Tracking Method of a Flying-Wing UAV Using the Differential Geometry

International Journal of Aerospace Engineering ◽

10.1155/2016/3406256 ◽

2016 ◽

Vol 2016 ◽

pp. 1-9 ◽

Cited By ~ 3

Author(s):

Yi Zhu ◽

Xin Chen ◽

Chuntao Li

Keyword(s):

Flight Control ◽

Trajectory Tracking ◽

Dimensional Space ◽

Three Dimensional ◽

Elliptic Cylinder ◽

Dynamic Equations ◽

Moving Frame ◽

Moving Frames ◽

Tracking Method ◽

Tracking Process

The problem of UAV trajectory tracking is a difficult issue for scholars and engineers, especially when the target curve is a complex curve in the three-dimensional space. In this paper, the coordinate frames during the tracking process are transformed to improve the tracking result. Firstly, the basic concepts of the moving frame are given. Secondly the transfer principles of various moving frames are formulated and the Bishop frame is selected as a final choice for its flexibility. Thirdly, the detailed dynamic equations of the moving frame tracking method are formulated. In simulation, a moving frame of an elliptic cylinder helix is formulated precisely. Then, the devised tracking method on the basis of the dynamic equations is tested in a complete flight control system with 6 DOF nonlinear equations of the UAV. The simulation result shows a satisfactory trajectory tracking performance so that the effectiveness and efficiency of the devised tracking method is proved.

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Minimal Surfaces by Moving Frames

Springer Collected Works in Mathematics - Selected Papers IV ◽

10.1007/978-1-4614-9451-5_8 ◽

1989 ◽

pp. 65-89

Author(s):

Shiing-Shen Chern

Keyword(s):

Minimal Surfaces ◽

Moving Frames

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Moving Frames and Coframes

Algebraic Methods in Physics ◽

10.1007/978-1-4613-0119-6_4 ◽

2001 ◽

pp. 47-64 ◽

Cited By ~ 4

Author(s):

Mark Fels ◽

Peter J. Olver

Keyword(s):

Moving Frames

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Stabilizing Ship Motion With a Dual System Inertial Disk

Volume 4: Dynamics, Vibration, and Control ◽

10.1115/imece2019-10250 ◽

2019 ◽

Author(s):

Torstein R. Storaas ◽

Kasper Virkesdal ◽

Gitle S. Brekke ◽

Thorstein Rykkje ◽

Thomas Impelluso

Keyword(s):

Oil And Gas ◽

New Technologies ◽

Equations Of Motion ◽

Moving Frame ◽

Moving Frames ◽

New Approach ◽

Lie Group Theory ◽

Moving Frame Method ◽

Frame Method ◽

Modern Mathematics

Abstract Norwegian industries are constantly assessing new technologies and methods for more efficient and safer maintenance in the aqua cultural, renewable energy, and oil and gas industries. These Norwegian offshore industries share a common challenge: to install new equipment and transport personnel in a safe and controllable way between ships, farms and platforms. This paper deploys the Moving Frame Method (MFM) to analyze ship stability moderated by a dual gyroscopic inertial device. The MFM describes the dynamics of the system using modern mathematics. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. This, together with a restriction on the variation of the angular velocity used in Hamilton’s principle, enables an effective way of extracting the equations of motion. This project extends previous work. It accounts for the dual effect of two inertial disk devices, it accounts for the prescribed spin of the disks. It separates out the prescribed variables. This work displays the results in 3D on cell phones. It represents a prelude to testing in a wave tank.

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Modeling a Knuckle-Boom Crane Control to Reduce Pendulum Motion Using the Moving Frame Method

Volume 4: Dynamics, Vibration, and Control ◽

10.1115/imece2019-10436 ◽

2019 ◽

Author(s):

Maren Eriksen Eia ◽

Elise Mari Vigre ◽

Thorstein Ravneberg Rykkje

Keyword(s):

Equations Of Motion ◽

Moving Frame ◽

Web Pages ◽

Moving Frames ◽

Pendulum Motion ◽

Moving Frame Method ◽

Frame Method ◽

The Masses ◽

And Control ◽

Boom Crane

Abstract A Knuckle Boom Crane is a pedestal-mounted, slew-bearing crane with a joint in the middle of the distal arm; i.e. boom. This distal boom articulates at the ‘knuckle (i.e.: joint)’ and that allows it to fold back like a finger. This is an ideal configuration for a crane on a ship where storage space is a premium. This project researches the motion and control of a ship mounted knuckle boom crane to minimize the pendulum motion of a hanging load. To do this, the project leverages the Moving Frame Method (MFM). The MFM draws upon Lie group theory — SO(3) and SE(3) — and Cartan’s Moving Frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. The work reported here accounts for the masses and geometry of all components, interactive motor couples and prepares for buoyancy forces and added mass on the ship. The equations of motion are solved numerically using a 4th order Runge Kutta (RK4), while solving for the rotation matrix for the ship using the Cayley-Hamilton theorem and Rodriguez’s formula for each timestep. This work displays the motion on 3D web pages, viewable on mobile devices.

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