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.

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 Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

scholarly journals Stabilization of a Rigid Body Payload With Multiple Cooperative Quadrotors

2016 ◽  
Vol 138 (12) ◽  
Author(s):  
Farhad A. Goodarzi ◽  
Taeyoung Lee
Keyword(s):  
Rigid Body ◽  
Arbitrary Number ◽  
Equations Of Motion ◽  
Vertical Direction ◽  
Free Form ◽  
Nonlinear Controller ◽  
Dynamics And Control ◽  
And Control ◽  
Flexible Cables ◽  
Quadrotor Unmanned Aerial Vehicles

Abstract This paper presents the full dynamics and control of arbitrary number of quadrotor unmanned aerial vehicles (UAVs) transporting a rigid body. The rigid body is connected to the quadrotors via flexible cables where each flexible cable is modeled as a system of arbitrary number of serially connected links. It is shown that a coordinate-free form of equations of motion can be derived for the complete model without any simplicity assumptions that commonly appear in other literature, according to Lagrangian mechanics on a manifold. A geometric nonlinear controller is presented to transport the rigid body to a fixed desired position while aligning all of the links along the vertical direction. A rigorous mathematical stability proof is given and the desirable features of the proposed controller are illustrated by numerical examples and experimental results.

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.

Nonlinear Dynamics and Control of an Ideal/Nonideal Load Transportation System With Periodic Coefficients

2006 ◽  
Vol 2 (1) ◽  
pp. 32-39 ◽  
Author(s):  
N. J. Peruzzi ◽  
J. M. Balthazar ◽  
B. R. Pontes ◽  
R. M. L. R. F. Brasil
Keyword(s):  
Equations Of Motion ◽  
Periodic Problem ◽  
Dc Motor ◽  
Transportation System ◽  
Stability Diagrams ◽  
Dynamics And Control ◽  
The Stability ◽  
Load Transportation ◽  
And Control ◽  
The Mathematical Model

In this paper, a loads transportation system in platforms or suspended by cables is considered. It is a monorail device and is modeled as an inverted pendulum built on a car driven by a dc motor. The governing equations of motion were derived via Lagrange’s equations. In the mathematical model we consider the interaction between the dc motor and the dynamical system, that is, we have a so called nonideal periodic problem. The problem is analyzed, qualitatively, through the comparison of the stability diagrams, numerically obtained, for several motor torque constants. Furthermore, we also analyze the problem quantitatively using the Floquet multipliers technique. Finally, we devise a control for the studied nonideal problem. The method that was used for analysis and control of this nonideal periodic system is based on the Chebyshev polynomial expansion, the Picard iterative method, and the Lyapunov-Floquet transformation (L-F transformation). We call it Sinha’s theory.

Dynamics and control of capture of a floating rigid body by a spacecraft robotic arm

2014 ◽  
Vol 33 (3) ◽  
pp. 315-332 ◽  
Author(s):  
Xiao-Feng Liu ◽  
Hai-Quan Li ◽  
Yi-Jun Chen ◽  
Guo-Ping Cai ◽  
Xi Wang
Keyword(s):  
Rigid Body ◽  
Robotic Arm ◽  
Dynamics And Control ◽  
And Control

Dynamics of Elastic Manipulators With Prismatic Joints

1992 ◽  
Vol 114 (1) ◽  
pp. 41-49 ◽  
Author(s):  
K. W. Buffinton
Keyword(s):  
Equations Of Motion ◽  
Elastic Beam ◽  
Alternative Form ◽  
Specific System ◽  
Beam Equations ◽  
Prismatic Joints ◽  
Dynamics And Control ◽  
And Control ◽  
Two Degree Of Freedom ◽  
Elastic Beam Equations

The purpose of this investigation is to study the formulation of equations of motion for flexible robots containing translationally moving elastic members that traverse a finite number of distinct support points. The specific system investigated is a two-degree-of-freedom manipulator whose configuration is similar to that of the Stanford Arm and whose translational member is regarded as an elastic beam. Equations of motion are formulated by treating the beam’s supports as kinematical constraints imposed on an unrestrained beam, by discretizing the beam by means of the assumed modes technique, and by applying an alternative form of Kane’s method which is particularly well suited for systems subject to constraints. The resulting equations are programmed and are used to simulate the system’s response when it performs tracking maneuvers. The results provide insights into some of the issues and problems involved in the dynamics and control of manipulators containing highly elastic members connected by prismatic joints.

Equations of Motion for Nonholonomic, Constrained Dynamical Systems via Gauss’s Principle

1993 ◽  
Vol 60 (3) ◽  
pp. 662-668 ◽  
Author(s):  
R. E. Kalaba ◽  
F. E. Udwadia
Keyword(s):  
Dynamical Systems ◽  
Closed Form ◽  
Programming Problem ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Closed Form Solution ◽  
Nonholonomic Constraints ◽  
Form Solution ◽  
Constrained Motion ◽  
Constraint Forces

In this paper we develop an analytical set of equations to describe the motion of discrete dynamical systems subjected to holonomic and/or nonholonomic Pfaffian equality constraints. These equations are obtained by using Gauss’s Principle to recast the problem of the constrained motion of dynamical systems in the form of a quadratic programming problem. The closed-form solution to this programming problem then explicitly yields the equations that describe the time evolution of constrained linear and nonlinear mechanical systems. The direct approach used here does not require the use of any Lagrange multipliers, and the resulting equations are expressed in terms of two different classes of generalized inverses—the first class pertinent to the constraints, the second to the dynamics of the motion. These equations can be numerically solved using any of the standard numerical techniques for solving differential equations. A closed-form analytical expression for the constraint forces required for a given mechanical system to satisfy a specific set of nonholonomic constraints is also provided. An example dealing with the position tracking control of a nonlinear system shows the power of the analytical results and provides new insights into application areas such as robotics, and the control of structural and mechanical systems.

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