Rigid Body In Terms Research Articles

Rolling Cones, Closed Attitude Trajectories, and Attitude Reconstruction

Louis Poinsot’s result, that any relative rotational motion between two frames can be realized as the motion of a moving cone rolling without slipping on a stationary cone, is stated and proved using matrix-vector algebra. Poinsot’s result is used to obtain a characterization of all attitude trajectories that are closed in the sense that the trajectory terminates at the same point that it starts from. The characterization is an extension to the continuous case of results on discrete closed sequences of rotations, and yields examples of closed attitude (orientation and angular velocity vector) trajectories that may be seen as extensions to the continuous-time case of classical theorems on discrete rotation sequences by Rodrigues, Hamilton and Donkin. One of the examples of closed attitude trajectories is used to obtain an extension of the Goodman–Robinson theorem, which reconstructs the instantaneous orientation of a rotating frame from the components of a fixed vector taken along this frame. This extension yields an explicit expression for the rotation matrix representing the instantaneous orientation of a torque-free rigid body in terms of the body components of its angular velocity.

Solution of the Forward Dynamics of a Single-loop Linkage Using Power Series

The kinematic differential equations express the paths taken by points, lines, and coordinate frames attached to a rigid body in terms of the instantaneous screw for the motion of that body. Such differential equations are linear but with a time-varying coefficient and hence solvable by power series. A single-loop kinematic chain may be expressed by a system of such differential equations subject to a linear constraint. A single matrix factorization followed by a sequence of substitutions of linear-system right-hand-side terms determines successive orders of the joint rate coefficients in the kinematic solution for this mechanism. The present work extends this procedure to the forward dynamics problem, applying it to a Clemens constant-velocity coupling expressed as a spatial 9R closed kinematic chain.

Series Solution for Finite Displacement of Planar Four-Bar Linkages

A time-varying instantaneous screw characterizes the motion of a rigid body. The kinematic differential equation expresses the path taken by any point on that rigid body in terms of this screw. Therefore, when a revolute joint is attached to a moving link in a planar kinematic chain, the path taken by the center of that revolute joint is the solution to such an equation. The instantaneous screw of a link in that chain is in turn determined by the action of the joints connecting that link to ground, where the contribution of each joint to that instantaneous screw is determined by its actuation rate and center point. Substituting power series expansions for joint rates into the kinematic differential equations for joint centers, and expressing loop closure as a linear constraint on the instantaneous screws of the links, a recurrence relation is established that solves for the coefficients in those power series. The resulting solution is applied to determine the equilibrium pendulum tilt of the United Aircraft TurboTrain. Comparing that power series approximation with an exact kinematic analysis shows convergence properties of the series.

An Alternative Derivation of the Quaternion Equations of Motion for Rigid-Body Rotational Dynamics

This note provides a direct method for obtaining Lagrange’s equations describing the rotational motion of a rigid body in terms of quaternions by using the so-called fundamental equation of constrained motion.

Equations of motion for general constrained systems in Lagrangian mechanics

This paper develops a new, simple, explicit equation of motion for general constrained mechanical systems that may have positive semi-definite mass matrices. This is done through the creation of an auxiliary mechanical system (derived from the actual system) that has a positive definite mass matrix and is subjected to the same set of constraints as the actual system. The acceleration of the actual system and the constraint force acting on it are then directly provided in closed form by the acceleration and the constraint force acting on the auxiliary system, which thus gives the equation of motion of the actual system. The results provide deeper insights into the fundamental character of constrained motion in general mechanical systems. The use of this new equation is illustrated through its application to the important and practical problem of finding the equation of motion for the rotational dynamics of a rigid body in terms of quaternions. This leads to a form for the equation describing rotational dynamics that has hereto been unavailable.

Interaction of Flows on Quadrics

We interpret the Rodrigues formula describing a torsion of a rigid body in terms of Lie derivatives. We consider a more general situation where vector fields on quadrics (ellipsoids and hyperboloids) are dragged by other vector fields. The question is about the adjoint representation of a Lie group which is more general than the usual torsion group of the three‐dimensional Euclidean space.

Further passivity results for the attitude control problem

In this paper we establish passivity for the system which describes the attitude motion of a rigid body in terms of minimal three-dimensional kinematic parameters. In particular, we show that linear asymptotically stabilizing controllers and control laws without angular velocity measurements follow naturally from these passivity properties. The results of this paper extend similar results for the case of the (nonminimal) Euler parameters.

Quantization of classical lagrangian mechanics

The equations of motion of a massive rigid body in the Lagrange—Poisson case (when one point of the body is fixed) are expressed in Hamiltonian form, making it possible to describe the Lagrangian rigid body in terms of classical mechanics. Using Berezin's quantization algorithm, it is possible to associate the Lagrangian classical mechanics with a quantum system, namely, a system of two-level particles interacting with a resonant field.

A note on the Kraichnan—Phillips theorem

The Kraichnan-Phillips theorem asserts that, for incompressible, homogeneous, turbulent flow over a plane, rigid wall, the wall-pressure wavenumber-frequency spectrum P(k, ω) → 0 as the planar wavenumber k → 0 provided the frequency ω ≠ 0. A proof of this theorem is given by use of a general formula that expresses the normal force on an arbitrary rigid body in terms of volume and surface integrals involving the vorticity. Implications for the theory of flow-induced surface vibrations are briefly discussed.

Anisotropic thermal-parameter refinement of the DNA dodecamer CGCGAATTCGCG by the segmented rigid-body method

A structure-factor least-squares refinement of the deoxyoligonucleotide (CGCGAATTCGCG)2 has been conducted using a model with constraints and restraints on the positional parameters and a segmented rigid-body representation for the anisotropic temperature factors. The macromolecule was divided into subgroups each of which was treated as a rigid body in terms of both positional and thermal parameters. For each subgroup, the thermal parameters determined were elements of translation, libration and correlation (TLS) tensors. This segmented rigidbody model of thermal motion has not previously been applied to the refinement of a macromolecular crystal structure. The anisotropic thermal-parameter refinement has significantly reduced the classical R factor as judged by the Hamilton test. The resulting difference Fourier map has a considerably lower noise level allowing fifteen additional low-occupancy water positions to be identified. In addition, analysis of the anisotropic thermal parameters has revealed new information about the local mobility of the groups in the oligonucleotide. Thus, the method of segmented rigid-body anisotropic temperature-factor refinement appears to be uniquely suited to macromolecules, especially nucleic acids, where high-resolution data are usually unavailable.

New integrable problem of classical mechanics

Complete integrability in Liouville's sense is proven for rotation of an arbitrary rigid body with a fixed point in a Newtonian field with an arbitrary homogeneous quadratic potential. A consequences is the complete integrability of rotation of a rigid body with fixed center of mass in the field of arbitrary sufficiently remote objects (in the second approximation). Explicit formulae are obtained expressing angular velocities of the rigid body in terms of θ-functions for Riemannian surfaces. Integrable cases are found for rotation of a rigid body in nonlinear Newtonian potential fields.

A Linear Algebra Approach to the Analysis of Rigid Body Motion from Position and Velocity Data

The location, orientation, and velocity of a rigid body can be defined in terms of three non-collinear points fixed in the body. As the rigid body moves through space, the displacement may be described by a series of rotations and translations and the velocity by the angular velocity of the body and the velocity of a point in the body. The displacement and velocity can be conveniently represented in matrix form by a series of matrices which describe the motion of the body between two finitely separated positions and two infinitesimally separated positions respectively.In this paper, solutions are presented which provide explicit formulas for the finite displacement of the rigid body in terms of the initial and final positions of three points fixed in the body and for the angular velocity and axis of rotation of the body in terms of the position and velocity of those same three points.

A Linear Algebra Approach to the Analysis of Rigid Body Displacement From Initial and Final Position Data

The location and orientation of a rigid body in space can be defined in terms of three noncollinear points in the body. As the rigid body is moved through space, the motion may be described by a series of rotations and translations. The sequence of displacements may be conveniently represented in matrix form by a series of displacement matrices that describe the motion of the body between successive positions. If the rotations and translations (and hence the displacement matrix) are known then succeeding positions of a rigid body may be easily calculated in terms of the initial position. Conversely, if successive positions of three points in the rigid body are known, it is possible to calculate the parameters of the corresponding rotation and translation. In this paper, a new solution is presented which provides explicit formulas for the rotation and translation of a rigid body in terms of the initial and final positions of three points fixed in the rigid body. The rotation matrix is determined directly whereupon appropriate rotation angles and other information can subsequently be calculated if desired.

A Quantum Theory of the Rigid Body in Terms of a Two-Component Spinor

A Quantum Theory of the Rigid Body in Terms of a Two-Component Spinor