Geometric Algorithms for Robot Dynamics: A Tutorial Review

2018 ◽  
Vol 70 (1) ◽  
Author(s):  
Frank C. Park ◽  
Beobkyoon Kim ◽  
Cheongjae Jang ◽  
Jisoo Hong
Keyword(s):  
Rigid Body ◽  
Inverse Dynamics ◽  
Group Structure ◽  
Equations Of Motion ◽  
Variable Stiffness ◽  
Rigid Bodies ◽  
Robot Dynamics ◽  
Motion Optimization ◽  
Geometric Formulation ◽  
Variable Stiffness Actuators

We provide a tutorial and review of the state-of-the-art in robot dynamics algorithms that rely on methods from differential geometry, particularly the theory of Lie groups. After reviewing the underlying Lie group structure of the rigid-body motions and the geometric formulation of the equations of motion for a single rigid body, we show how classical screw-theoretic concepts can be expressed in a reference frame-invariant way using Lie-theoretic concepts and derive recursive algorithms for the forward and inverse dynamics and their differentiation. These algorithms are extended to robots subject to closed-loop and other constraints, joints driven by variable stiffness actuators, and also to the modeling of contact between rigid bodies. We conclude with a demonstration of how the geometric formulations and algorithms can be effectively used for robot motion optimization.

Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices

1999 ◽  
Vol 66 (4) ◽  
pp. 986-996 ◽  
Author(s):  
S. K. Saha
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Kinematic Chain ◽  
Rigid Bodies ◽  
Dynamic Equations ◽  
Orthogonal Complement ◽  
Forward Dynamics ◽  
Velocity Constraints

Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n—n being the degrees-of-freedom of the system at hand—inverse dynamics and order n3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the system’s accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.

scholarly journals Dynamic model of multi-rigid-body systems based on particle dynamics with recursive approach

2005 ◽  
Vol 2005 (4) ◽  
pp. 365-382 ◽  
Author(s):  
Hazem Ali Attia
Keyword(s):  
Rigid Body ◽  
Dynamic Model ◽  
Equations Of Motion ◽  
Particle Dynamics ◽  
Rigid Bodies ◽  
Constrained System ◽  
Open Chain ◽  
Chain System ◽  
Closed Chain ◽  
Serial Chains

A dynamic model for multi-rigid-body systems which consists of interconnected rigid bodies based on particle dynamics and a recursive approach is presented. The method uses the concepts of linear and angular momentums to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotational transformation matrix. For the open-chain system, the equations of motion are generated recursively along the serial chains. A closed-chain system is transformed to open-chain by cutting suitable kinematical joints and introducing cut-joint constraints. An example is chosen to demonstrate the generality and simplicity of the developed formulation.

Vibration Modes and Frequencies of Timoshenko Beams With Attached Rigid Bodies

1995 ◽  
Vol 62 (1) ◽  
pp. 193-199 ◽  
Author(s):  
M. W. D. White ◽  
G. R. Heppler
Keyword(s):  
Boundary Conditions ◽  
Rigid Body ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Rigid Bodies ◽  
The Body ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
First Moment

The equations of motion and boundary conditions for a free-free Timoshenko beam with rigid bodies attached at the endpoints are derived. The natural boundary conditions, for an end that has an attached rigid body, that include the effects of the body mass, first moment of mass, and moment of inertia are included. The frequency equation for a free-free Timoshenko beam with rigid bodies attached at its ends which includes all the effects mentioned above is presented and given in terms of the fundamental frequency equations for Timoshenko beams that have no attached rigid bodies. It is shown how any support / rigid-body condition may be easily obtained by inspection from the reported frequency equation. The mode shapes and the orthogonality condition, which include the contribution of the rigid-body masses, first moments, and moments of inertia, are also developed. Finally, the effect of the first moment of the attached rigid bodies is considered in an illustrative example.

The Dual Inertia Operator and Its Application to Robot Dynamics

1994 ◽  
Vol 116 (4) ◽  
pp. 1089-1095 ◽  
Author(s):  
V. Brodsky ◽  
M. Shoham
Keyword(s):  
Rigid Body ◽  
Robot Manipulator ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Three Dimensional ◽  
Dynamic Equations ◽  
Robot Dynamics ◽  
Dual Vector ◽  
Vector Algebra ◽  
Vector Part

The principle of transference states that when dual numbers replace real ones all laws of vector algebra, which describe the kinematics of rigid body with one point fixed, are also valid for motor algebra, which describes a free rigid body. No such direct extension exists, however, for dynamics. Rather, the inertia binor is used to obtain the dual momentum, from which the dual equations of motion are derived. This raises the dual dynamic equations to six dimensions, and in fact, does not act on the dual vector as a whole, but on its real and dual parts as two distinct real vectors. Moreover, in order to obtain the dual force as a derivative of the dual momentum in a correct order, real and dual parts have to be artificially interchanged. In this investigation the dual inertia operator, which allows direct relation of dual momentum to dual velocity, is introduced. It gives the mass a dual property which has the inverse sense of Clifford’s dual unit, namely, it reduces a motor to a rotor proportional to the vector part of the former. In a way analogous to the principle of transference, the same equation of momentum and its time derivative, which holds for a linear motion, holds for both linear and angular motion of a rigid body if dual force, dual velocity, and dual inertia replace their real counterparts. It is shown that by systematic application of the dual inertia for derivation of the dual momentum and the dual energy, both Newton-Euler and Lagrange formulations of equations of motion are obtained in a complete three-dimensional dual form. As an example, these formulations are used to derive the inverse dual dynamic equations of a robot manipulator.

On the Euler–Lagrange Equation for Planar Systems of Rigid Bodies or Lumped Masses

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
R. Wiebe ◽  
P. S. Harvey
Keyword(s):  
Rigid Body ◽  
Reference Frames ◽  
Dynamic Equilibrium ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Double Pendulum ◽  
Governing Equations ◽  
Lumped Mass ◽  
Planar Systems

The Euler–Lagrange equation is frequently used to develop the governing dynamic equilibrium expressions for rigid-body or lumped-mass systems. In many cases, however, the rectangular coordinates are constrained, necessitating either the use of Lagrange multipliers or the introduction of generalized coordinates that are consistent with the kinematic constraints. For such cases, evaluating the derivatives needed to obtain the governing equations can become a very laborious process. Motivated by several relevant problems related to rigid-body structures under seismic motions, this paper focuses on extending the elegant equations of motion developed by Greenwood in the 1970s, for the special case of planar systems of rigid bodies, to include rigid-body rotations and accelerating reference frames. The derived form of the Euler–Lagrange equation is then demonstrated with two examples: the double pendulum and a rocking object on a double rolling isolation system. The work herein uses an approach that is used by many analysts to derive governing equations for planar systems in translating reference frames (in particular, ground motions), but effectively precalculates some of the important stages of the analysis. It is hoped that beyond re-emphasizing the work by Greenwood, the specific form developed herein may help researchers save a significant amount of time, reduce the potential for errors in the formulation of the equations of motion for dynamical systems, and help introduce more researchers to the Euler–Lagrange equation.

scholarly journals CASES OF INTEGRABILITY CORRESPONDING TO THE PENDULUM MOTION IN FOUR-DIMENSIONAL SPACE

2017 ◽  
Vol 23 (1) ◽  
pp. 41-58
Author(s):  
M. V. Shamolin
Keyword(s):  
Rigid Body ◽  
Force Fields ◽  
Characteristic Point ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
The Body ◽  
Pendulum Motion ◽  
Nonconservative Force ◽  
Similar Force

In this article, we systemize some results on the study of the equations of motion of dynamically symmetric fixed four-dimensional rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of the motion of a free four-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. We also show the nontrivial topological and mechanical analogies.

Modeling of Multibody Flexible Articulated Structures With Mutually Coupled Motions: Part I — General Theory

1992 ◽  
Author(s):  
Jiechi Xu ◽  
Joseph R. Baumgarten
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Local Level ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Moving Frame ◽  
Element Analysis ◽  
Highly Nonlinear ◽  
Geometric Boundary ◽  
System Equations

Abstract In the present paper a general systematic modeling procedure has been conducted in deriving dynamic equations of motion using Lagrange’s approach for a spatial multibody structural system involving rigid bodies and elastic members. Both the rigid body degrees of freedom and the elastic degrees of freedom are considered as unknown generalized coordinates of the entire system in order to reflect the nature of mutually coupled rigid body and elastic motions. The assumption of specified rigid body gross motion is no longer necessary in the equation derivation and the resulting differential equations are highly nonlinear. Finite element analysis (FEA) with direct stiffness method has been employed to model the flexible substructures. Nonlinear coupling terms between the rigid body and elastic motions are fully derived and are explicitly expressed in matrix form. The equations of motion of each individual subsystem are formulated based on a moving frame instead of a traditional inertial frame. These local level equations of motion are assembled to obtain the system equations with the implementation of geometric boundary conditions by means of a compatibility matrix.

Explicit Poincaré equations of motion for general constrained systems. Part II. Applications to multi-body dynamics and nonlinear control

2007 ◽  
Vol 463 (2082) ◽  
pp. 1435-1446 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Control ◽  
Rigid Body ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Constrained Systems ◽  
The Third ◽  
Highly Nonlinear ◽  
Body Dynamics ◽  
Multi Body

The power of the new equations of motion developed in part I of this paper is illustrated using three examples from multi-body dynamics. The first two examples deal with the problem of accurately controlling the orientation of a rigid body, while the third example deals with the synchronization of two rigid bodies so that their relative orientations are ‘locked’ through prescribed dynamical relationships. The ease, simplicity and accuracy with which control of such highly nonlinear systems is achieved are demonstrated.

Robust adaptive vibration control of an underactuated flexible spacecraft

2018 ◽  
Vol 25 (4) ◽  
pp. 834-850 ◽  
Author(s):  
H. MoradiMaryamnegari ◽  
A.M. Khoshnood
Keyword(s):  
Rigid Body ◽  
Vibration Control ◽  
Control Method ◽  
Frequency Estimation ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Lagrange Equations ◽  
Flexible Bodies ◽  
Robust Adaptive ◽  
Adaptive Vibration Control

Designing a controller for multi-body systems including flexible and rigid bodies has always been one of the major engineering challenges. Equations of motion of these systems comprise extremely nonlinear and coupled terms. Vibrations of flexible bodies affect sensors of rigid bodies and might make the system unstable. Introducing a new control strategy for designing control systems which do not require the rigid–flexible coupling model and can dwindle vibrations without sensors or actuators on flexible bodies is the purpose of this paper. In this study, a spacecraft comprising a rigid body and a flexible panel is used as the case study, and its equations of motion are extracted using Lagrange equations in terms of quasi-coordinates. For oscillations on a rigid body to be eliminated, a frequency estimation algorithm and an adaptive filtering are used. A controller is designed based on the rigid model of the system, and then robust stability conditions for the rigid–flexible system are obtained. The conditions are also developed for the spacecraft with more than one active frequency. Finally, the robust adaptive vibration control system is simulated in the presence of resonance. Simulations’ results indicate the advantage of the control method even when several active frequencies simultaneously resonate the dynamics system.

Experimental and Analytical Study of Free Lift-Off Motion Induced Slip Behavior of Rectangular Rigid Bodies

2004 ◽  
Vol 126 (1) ◽  
pp. 53-58 ◽  
Author(s):  
Tomoyo Taniguchi
Keyword(s):  
Energy Balance ◽  
Rigid Body ◽  
Variational Approach ◽  
Analytical Study ◽  
Mechanical Energy ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Time Histories ◽  
Analytical Accuracy ◽  
Lift Off

The mechanism of the slip of a rectangular rigid body during free lift-off motion is investigated analytically and experimentally. Equations of motion of slipping rectangular rigid body during free lift-off motion are derived by the variational approach. The time histories of both slip and lift-off motions are numerically computed and compared with corresponding experimental results to discuss the analytical accuracy; given an initial enforced lift-off angle to the rectangular rigid body and then gently released. The mechanical energy balance of simultaneous slip and lift-off motions is clarified to explain the considerable reduction of lift-off angle due to the slip.

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