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

2010 ◽  
Vol 77 (4) ◽  
Author(s):  
Firdaus E. Udwadia ◽  
Aaron D. Schutte
Keyword(s):  
Rigid Body ◽  
Rotational Motion ◽  
Direct Method ◽  
Equations Of Motion ◽  
Fundamental Equation ◽  
Rotational Dynamics ◽  
Constrained Motion ◽  
Alternative Derivation ◽  
A Direct Method ◽  
Lagrange's Equations

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.

scholarly journals A unified approach to rigid body rotational dynamics and control

2011 ◽  
Vol 468 (2138) ◽  
pp. 395-414 ◽  
Author(s):  
Firdaus E. Udwadia ◽  
Aaron D. Schutte
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Equations Of Motion ◽  
Fundamental Equation ◽  
Rotational Dynamics ◽  
Constrained Motion ◽  
Dynamics And Control ◽  
Common Framework ◽  
The Stability ◽  
And Control

This paper develops a unified methodology for obtaining both the general equations of motion describing the rotational dynamics of a rigid body using quaternions as well as its control. This is achieved in a simple systematic manner using the so-called fundamental equation of constrained motion that permits both the dynamics and the control to be placed within a common framework. It is shown that a first application of this equation yields, in closed form, the equations of rotational dynamics, whereas a second application of the self-same equation yields two new methods for explicitly determining, in closed form, the nonlinear control torque needed to change the orientation of a rigid body. The stability of the controllers developed is analysed, and numerical examples showing the ease and efficacy of the unified methodology are provided.

Explicit Nonlinear Rotational Controllers Using the Fundamental Equation

2009 ◽  
Author(s):  
Aaron D. Schutte ◽  
Firdaus E. Udwadia
Keyword(s):  
Asymptotic Stability ◽  
Rigid Body ◽  
Fundamental Equation ◽  
Rigid Bodies ◽  
Rotational Dynamics ◽  
Qualitative Behavior ◽  
Constrained Motion ◽  
Numerical Examples ◽  
Nonlinear Controllers ◽  
Quaternion Space

In this paper, we present two explicitly generated nonlinear controllers for rest-to-rest rigid body rotational maneuvers in terms of quaternions. The controllers are brought about by applying the fundamental equation of constrained motion to both the rotational dynamics and rotational control of rigid bodies. The first controller yields asymptotic stability at a desired orientation while allowing the stabilization to occur exactly along a pre-selected trajectory for three of the four components that make-up the quaternion. The second controller provides global stability at the desired orientation allowing stable motion to occur from any point in quaternion space. Numerical examples are provided showing the qualitative behavior that both rotational controllers yield when applied to a rigid body.

A Variationally Consistent Approach to Constrained Motion

2016 ◽  
Vol 83 (5) ◽  
Author(s):  
John T. Foster
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
A Priori ◽  
Nonholonomic Systems ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Constrained Motion ◽  
Consistent Approach ◽  
Holonomic And Nonholonomic Systems ◽  
D'alembert's Principle

A variationally consistent approach to constrained rigid-body motion is presented that extends D'Alembert's principle in a way that has a form similar to Kane's equations. The method results in minimal equations of motion for both holonomic and nonholonomic systems without a priori consideration of preferential coordinates.

A direct method for determining meshing surfaces of globoidal cams with tapered roller followers

2009 ◽  
Vol 223 (7) ◽  
pp. 1667-1676 ◽  
Author(s):  
D M Tsay ◽  
H P Chen ◽  
S Y Lin
Keyword(s):  
Rigid Body ◽  
Direct Method ◽  
Simple Approach ◽  
Numerical Results ◽  
Traditional Methods ◽  
Convenient Tool ◽  
Cam Profile ◽  
Tapered Roller ◽  
Rigid Body Transformation ◽  
A Direct Method

For determining the globoidal cam profile with tapered rollers radially mounted at its turret, a direct and relative simple approach is presented. The concept of the proposed procedure is based on the cam surfaces that can be represented as the swept surfaces of the tapered roller of the globoidal cam-turret mechanism in motion. For finding the swept surfaces of the tapered roller, the pitch surface yielded by the locus of the tapered roller is first generated through the rigid body transformation. Then, in the process of producing the swept surfaces for the cam surfaces, the meshing angles and meshing vectors are defined and identified by using the pitch surface of the tapered roller. Since it is not necessary to solve any non-linear contact equations usually encountered in traditional methods, such a procedure provides a convenient tool in determining the spatial cam surfaces. To demonstrate the feasibility and accuracy of the proposed approach, its analytical formulations with numerical results are compared with those obtained by an earlier technique based on the theory of screws.

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