We present the three main mathematical constructs used to represent the attitude of a rigid body in threedimensional space. These are (1) the rotation matrix, (2) a triple of Euler angles, and (3) the unit quaternion. To these we add a fourth, the rotation vector, which has many of the benefits of both Euler angles and quaternions, but neither the singularities of the former, nor the quadratic ...

It is shown in this paper that Euler was first to derive the finite rotation formula which is often erroneously attributed to Rodrigues, while Rodrigues was responsible for the derivation of the composition formulae for successive finite rotations and the so-called Euler parameters of finite rotation. Therefore, based upon historical facts, the following nomenclature is suggested: Euler's finit...

Let G be a finite group. The symmetric Euler characteristic χ(G) is the maximal Euler characteristic of any surface X (orientable or nonorientable) on which G acts. The groups of symmetric Euler characteristic χ ≥ −2 have been classified. We show that S5 is the unique group with Euler characteristic -3. Two related parameters are the symmetric genus σ and the symmetric crosscap number σ̃. We con...

Sieve trays are widely used in the gas- liquid contactors such as distillation and absorption towers. In this article, a three-dimensional, two phase CFD model using Euler-Euler framework was developed to simulate a distillation tower with two sieve trays. Hydrodynamic simulation of air and water system in different rates of gas phase was carried out and velocity distribution parameters, clear ...

Gregory S. Chirikjian Johns Hopkins University 2.1 Rotations in Three Dimensions Rules for Composing Rotations • Euler Angles • The Matrix Exponential 2.2 Full Rigid-Body Motion Composition of Motions • Screw Motions 2.3 Homogeneous Transforms and the Denavit-Hartenberg Parameters Homogeneous Transformation Matrices • The Denavit-Hartenberg Parameters in Robotics 2.4 Infinitesimal Motions and A...

A state feedback H 1 suboptimal control problem on T SO(3) for a rigid spacecraft with three control torques and disturbances is addressed. The Hamilton-Jacobi inequality associated with a corresponding state feedback H 1 suboptimal control problem on T S 3 is solved globally. The H 1 suboptimal state feedback on T S 3 involves the Euler parameters and solves the original problem on T SO(3) whe...

Sieve trays are widely used in the gas- liquid contactors such as distillation and absorption towers. In this article, a three-dimensional, two phase CFD model using Euler-Euler framework was developed to simulate a distillation tower with two sieve trays. Hydrodynamic simulation of air and water system in different rates of gas phase was carried out and velocity distribution parameters, clear ...

Many models of three-dimensional rigid body dynamics employ Euler parameters as rotational coordinates. Since the four Euler parameters are not independent, one has to consider the quaternion constraint in the equations of motion. This is usually done by the Lagrange multiplier technique. In the present paper, various forms of the rotational equations of motion will be derived, and it will be s...

Abstract: First, a brief overview is provided on humanoid robots, and also models for the dynamic behavior are discussed. As base for these models these two methods Denavit-Hartenberg and Newton-Euler are used. Main aim of this work is to investigate the stability of a biped robot developed from IHRT. There is currently the low base of robot consisting of feet, legs, hips and upper part of robo...