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 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.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

Geometric Insight Into the Dynamics of a Rigid Body Using the Spatial Triangle of Screws

2001 ◽  
Author(s):  
Gordon R. Pennock ◽  
Patrick J. Meehan
Keyword(s):  
Rigid Body ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Analytical Techniques ◽  
Rate Of Change ◽  
Time Rate ◽  
Open Chain ◽  
Spatial Mechanisms ◽  
Kinetics Of ◽  
Insight Into

Abstract Geometric relationships between the velocity screw and momentum screw are presented, and the dual angle between these two screws is shown to provide important insight into the kinetics of a rigid body. Then the centripetal screw is defined, and the significance of this screw in a study of the dynamics of a rigid body is explained. The dual-Euler equation, which is the dual form of the Newton-Euler equations of motion, is shown to be a spatial triangle. The vertices of the triangle are the centripetal screw, the time rate of change of momentum screw, and the force screw. The sides of the triangle are three dual angles between the three vertices. The spatial triangle provides valuable geometrical insight into the dynamics of a rigid body and is believed to be a meaningful alternative to existing analytical techniques. The authors believe that the work presented in this paper will prove useful in a dynamic analysis of closed-loop spatial mechanisms and multi-rigid body open-chain systems.

Evolution of Rotations of a Rigid Body Under the Action of Restoring and Control Moments

2005 ◽  
Author(s):  
L. D. Akulenko ◽  
D. D. Leshchenko ◽  
T. A. Kozachenko
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
The Body ◽  
System Of Equations ◽  
Slow Time ◽  
Case Examples ◽  
First Approximation ◽  
Nutation Angle ◽  
Regular Precession ◽  
And Control

Perturbed rotations of a rigid body close to the regular precession in the Lagrangian case under the action of a restoring moment depending on slow time and nutation angle, as well as a perturbing moment slowly varying with time, are studied. The body is assumed to spin rapidly, and the restoring and perturbing moments are assumed to be small with a certain hierarchy of smallness of the components. A first approximation averaged system of equations of motion for an essentially nonlinear two-frequency system is obtained in the nonresonance case. Examples of motion of a body under the action of particular restoring, perturbing, and control moments of force are considered.

Comparison of Selected Methods of Handling Redundant Constraints in Multibody Systems Simulations

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Marek Wojtyra ◽  
Janusz Frączek
Keyword(s):  
Numerical Methods ◽  
Rigid Body ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Point Of View ◽  
Redundant Constraints ◽  
Reaction Solution ◽  
Constraints Handling ◽  
Mathematical Operations ◽  
Augmented Lagrangian Formulation

When redundant constraints are present in a rigid body mechanism, only selected (if any at all) joint reactions can be determined uniquely, whereas others cannot. Analytic criteria and numerical methods of finding joints with uniquely solvable reactions are available. In this paper, the problem of joint reactions solvability is examined from the point of view of selected numerical methods frequently used for handling redundant constraints in practical simulations. Three different approaches are investigated in the paper: elimination of redundant constraints; pseudoinverse-based calculations; and the augmented Lagrangian formulation. Each method is briefly summarized; the discussion is focused on techniques of handling redundant constraints and on joint reactions calculation. In the case of multibody systems with redundant constraints, the rigid body equations of motion are insufficient to calculate some or all joint reactions. Thus, purely mathematical operations are performed in order to find the reaction solution. In each investigated method, the redundant constraints are treated differently, which—in the case of joints with nonunique reactions—leads to different reaction solutions. As a consequence, reactions reflecting the redundancy handling method rather than physics of the system are calculated. A simple example of each method usage is presented, and calculated joint reactions are examined. The paper points out the origins of nonuniqueness of constraint reactions in each examined approach. Moreover, it is shown that one and the same method may lead to different reaction solutions, provided that input data are prepared differently. Finally, it is demonstrated that—in case of joints with solvable reactions—the obtained solutions are unique, regardless of the method used for redundant constraints handling.

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.

Continuum of Motion Equations and Control Laws for the Inverted Pendulum Cart and Rotary Pendulum

2020 ◽  
Author(s):  
Constance Lare ◽  
Warren N. White
Keyword(s):  
Inverted Pendulum ◽  
Equations Of Motion ◽  
Control Law ◽  
Control Laws ◽  
Motion Equations ◽  
Base Radius ◽  
Limiting Condition ◽  
And Control ◽  
The Given ◽  
Insight Into

Abstract This paper questions whether the controller properties for a given rigid body mechanical system still apply as the given system is changed. As a first attempt in this investigation, the controller for the underactuated rotary pendulum is investigated as the system morphs into an underactuated inverted pendulum cart. As the limiting condition of the inverted pendulum cart is approached, the investigation allows the controller to also morph. The authors show that, as the pendulum base radius grows, the rotary pendulum equations of motion morph into the inverted pendulum cart dynamics. The paper presents necessary conditions for the successful morphing of the dynamic equations. The morphing process for the controller tests the idea whether the control law also satisfies the same continuum basis as the motion equations. The paper presents a framework for the class of controllers investigated for providing insight into when the controller morphing may be successful. This paper presents dimensionless quantities that render the equations of motion and controller for the inverted pendulum cart and rotary pendulum into dimensionless form. These dimensionless quantities allow comparison of controllers and systems that are not possible through simple inspection. This comparison ability is especially useful for quantifying the nonlinearities of a given system and controller compared to another system and controller having different parameter sizes, a comparison rarely seen in the control literature.

Modeling a Robot With Flexible Joints and Decoupling its Equations of Motion

1997 ◽  
Author(s):  
Ali Meghdari ◽  
Farbod Fahimi
Keyword(s):  
Rigid Body ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Flexible Joints ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Recent Advances ◽  
Dynamics Of Multibody Systems ◽  
Two Degree Of Freedom

Abstract Recent advances in the study of dynamics of multibody systems indicate the need for decoupling of the equations of motion. In this paper, our efforts are focused on this issue, and we have tried to expand the existing methods for multi-rigid body systems to include systems with some kind of flexibility. In this regard, the equations of motion for a planar two-degree-of-freedom robot with flexible joints is carried out using Lagrange’s equations and Kane’s equations with congruency transformations. The method of decoupling the equations of motion using Kane’s equations with congruency transformations is presented. Finally, the results obtained from both methods are compared.

Modeling and Control of Transverse and Torsional Vibrations in a Spherical Robotic Manipulator: Theoretical and Experimental Results

1997 ◽  
Vol 119 (3) ◽  
pp. 421-430 ◽  
Author(s):  
Liming Chen ◽  
Nabil G. Chalhoub
Keyword(s):  
Rigid Body ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Robotic Manipulator ◽  
Gain Scheduling ◽  
Torsional Vibrations ◽  
Flexible Link ◽  
Modeling And Control ◽  
Prismatic Joint ◽  
And Control

The present work addresses modeling and control issues pertaining to the positioning and orientating of rigid body payloads as they are being manipulated by flexible spherical robotic manipulators. A general approach, to systematically derive the equations of motion of the robotic manipulator, is used herein. The objective of the controller is to yield a desired rigid body response of the arm while damping out the transverse and torsional vibrations of the compliant link. Note that the control objective has to be achieved by solely relying on the existing joint actuators whose band-widths are far below the natural frequencies of the torsional modes. The current work demonstrates that, in spite of the physical limitations of the system, the controller can actively damp out the torsional vibrations by relying on the coupling terms between the torsional vibrations and the remaining degrees of freedom of the arm. Moreover, a gain scheduling procedure is introduced to continuously tune the controller to the natural frequencies of the flexible link whose length is varied by the prismatic joint. The digital simulation results demonstrate the capability of the “rigid and flexible motion controller (RFMC)” in drastically attenuating the transverse and torsional vibrations during point-to-point (PTP) maneuvers of the arm. Furthermore, the gain scheduling procedure is shown to significantly reduce the degradations in the RFMC performance that are brought about by having the flexible link connected to a prismatic joint. A limited experimental work has also been conducted to demonstrate the viability of the proposed approach.

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