A New O(n) Method for Inverting the Mass Matrix for Serial Chains Composed of Rigid Bodies

2005 ◽  
Author(s):  
Yunfeng Wang ◽  
Gregory S. Chirikjian
Keyword(s):  
Multibody Dynamics ◽  
Inverse Dynamics ◽  
Mass Matrix ◽  
Rotation Group ◽  
Rigid Bodies ◽  
Euclidean Group ◽  
The Past ◽  
Lumped Masses ◽  
Serial Chains ◽  
And Robotics

Over the past several decades a number of O(n) methods for forward and inverse dynamics computations have been developed in the multibody dynamics and robotics literature. In this paper, a method developed in 1973 by Fixman for O(n) computation of the mass-matrix determinant for a polymer chain consisting of point masses is adapted and modified. In other recent papers, we and our collaborators recently extended this method in order for Fixman’s results to be applicable to robotic manipulator models with lumped masses. In the present paper we extend these ideas further to the case of serial chains composed of rigid-bodies. This requires the use of relatively deep mathematics associated with the rotation group, SO(3), and the special Euclidean group, SE(3), and how to differentiate functions of group-valued argument.

scholarly journals O(n) mass matrix inversion for serial manipulators and polypeptide chains using Lie derivatives

Robotica ◽  
2007 ◽  
Vol 25 (6) ◽  
pp. 739-750 ◽  
Author(s):  
Kiju Lee ◽  
Yunfeng Wang ◽  
Gregory S. Chirikjian
Keyword(s):  
Inverse Dynamics ◽  
Mass Matrix ◽  
Rotation Group ◽  
Rigid Bodies ◽  
Euclidean Group ◽  
Serial Manipulators ◽  
The Past ◽  
Polypeptide Chains ◽  
Serial Chains ◽  
And Robotics

SUMMARYOver the past several decades, a number of O(n) methods for forward and inverse dynamics computations have been developed in the multibody dynamics and robotics literature. A method was developed by Fixman in 1974 for O(n) computation of the mass-matrix determinant for a serial polymer chain consisting of point masses. In other of our recent papers, we extended this method in order to compute the inverse of the mass matrix for serial chains consisting of point masses. In the present paper, we extend these ideas further and address the case of serial chains composed of rigid-bodies. This requires the use of relatively deep mathematics associated with the rotation group, SO(3), and the special Euclidean group, SE(3), and specifically, it requires that one differentiates real-valued functions of Lie-group-valued argument.

Instantaneous impulses in multibody dynamics

2015 ◽  
Vol 22 (4) ◽  
pp. 581-635
Author(s):  
P Lidström
Keyword(s):  
Multibody Dynamics ◽  
Virtual Work ◽  
Mass Matrix ◽  
Full Rank ◽  
Rigid Bodies ◽  
Unilateral Constraints ◽  
Lagrange’S Equations ◽  
Bilateral Constraints ◽  
The Impact ◽  
Lagrange's Equations

This paper considers instantaneous impulses in multibody dynamics. Instantaneous impulses may act on the multibody from its exterior or they may appear in its interior as a consequence of two of its parts interacting by an impact imposed by a unilateral constraint. The theory is based on the Euler laws of instantaneous impulses, which may be seen as a complement to the Euler laws for regular motions. Based on these laws, and specific continuum properties of the quantities involved, local balance laws for momentum and moment of momentum, involving instantaneous impulses and introducing the Cauchy impulse tensor, are derived. Thermodynamical restrictions on the impulse tensor are formulated based on the dissipation inequality. By stating a principle of virtual work for instantaneous impulses, and demonstrating its equivalence to Euler’s laws, Lagrange’s equations are derived. Lagrange’s equations are convenient to use in the case of multibody dynamics containing rigid as well as flexible parts. A central theme of this paper is the discussion of the interaction between parts of the multibody and their relation to geometrical and kinematical constraints. This interaction is severely affected by the presence of friction, which is notoriously difficult to handle. In a preparation for this discussion we first consider the one-point impact between two rigid bodies. The importance of the so-called impact tensor for this problem is demonstrated. In order to be able to handle the impact laws of Poisson and Stonge, an impact process, governed by a system of ordinary differential equations, is defined. Within this model phenomena, such as slip stop, slip start and slip direction reversal, may be handled. For a multibody with an arbitrary number of parts and multiple impacts, the situation is much more complicated and certain simplifications have to be introduced. Equations of motion for a multibody, consisting of rigid parts and in the presence of ideal bilateral constraints and unilateral constraints involving friction, are formulated. Unique solutions are obtained, granted that the mass matrix of the multibody system is non-singular, the constraint matrices satisfy specific full rank conditions and that the friction is not too high.

Flexible Multibody Dynamics Formulation by Hamilton’s Equations

2010 ◽  
Author(s):  
Martin M. Tong
Keyword(s):  
Multibody Dynamics ◽  
Multibody System ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Flexible Body ◽  
Flexible Multibody System ◽  
Generalized Coordinates ◽  
Flexible Multibody ◽  
The Matrix

Numerical solution of the dynamics equations of a flexible multibody system as represented by Hamilton’s canonical equations requires that its generalized velocities q˙ be solved from the generalized momenta p. The relation between them is p = J(q)q˙, where J is the system mass matrix and q is the generalized coordinates. This paper presents the dynamics equations for a generic flexible multibody system as represented by p˙ and gives emphasis to a systematic way of constructing the matrix J for solving q˙. The mass matrix is shown to be separable into four submatrices Jrr, Jrf, Jfr and Jff relating the joint momenta and flexible body mementa to the joint coordinate rates and the flexible body deformation coordinate rates. Explicit formulas are given for these submatrices. The equations of motion presented here lend insight to the structure of the flexible multibody dynamics equations. They are also a versatile alternative to the acceleration-based dynamics equations for modeling mechanical systems.

scholarly journals A Simulation Model for Computing the Loads Generated at Landing Site During Helicopter Take-Off or Landing Operation

2019 ◽  
Vol 2019 (2) ◽  
pp. 59-75
Author(s):  
Jarosław Stanisławski
Keyword(s):  
Equations Of Motion ◽  
Landing Site ◽  
Simulation Method ◽  
Rigid Bodies ◽  
Landing Gear ◽  
Rotor Blades ◽  
Wind Gust ◽  
The Galerkin Method ◽  
Lumped Masses ◽  
And Control

Summary The paper presents simulation method and results of calculations determining behavior of helicopter and landing site loads which are generated during phase of the helicopter take-off and landing. For helicopter with whirling rotor standing on ground or touching it, the loads of landing gear depend on the parameters of helicopter movement, occurrence of wind gusts and control of pitch angle of the rotor blades. The considered model of helicopter consists of the fuselage and main transmission treated as rigid bodies connected with elastic elements. The fuselage is supported by landing gear modeled by units of spring and damping elements. The rotor blades are modeled as elastic axes with sets of lumped masses of blade segments distributed along them. The Runge-Kutta method was used to solve the equations of motion of the helicopter model. According to the Galerkin method, it was assumed that the parameters of the elastic blade motion can be treated as a combination of its bending and torsion eigen modes. For calculations, data of a hypothetical light helicopter were applied. Simulation results were presented for the cases of landing helicopter touching ground with different vertical speed and for phase of take-off including influence of rotor speed changes, wind gust and control of blade pitch. The simulation method may help to define the limits of helicopter safe operation on the landing surfaces.

Vibrations Analysis for Gun Systems by Multibody Dynamics

1999 ◽  
Author(s):  
Deshi Wang ◽  
Renbin Xiao ◽  
Guangyao Ouyang
Keyword(s):  
Multibody Dynamics ◽  
Maximum Amplitude ◽  
Rigid Bodies ◽  
Lagrangian Equations ◽  
Structure Vibration

Abstract The multi-rigid-bodies model for the single barrel gun has been developed and the structure vibration of the gun has been analyzed though Lagrangian equations reduced by the software on microcomputer. It is shown that the deck stiffness is significant in the effect of the gun vibration, and then that design of a suitable equilibrium equipment and design of the elastic connection axle to increasing the flexibility in pitching movement can effectively decrease the vibration amplitudes of the tube end. The maximum amplitude, for example, could decay to ten percent of that when the measures mentioned above are adopted. It also indicates that entirely removing the vibration is impossible except relevant vibration absorbing measures are adopted. The method and results are also significant in checking the motions of the mechanisms designed in the guns.

A Comparison of Kinetostatic and Multibody Dynamics Models for Simulating Protein Structures

2007 ◽  
Author(s):  
Hai-Jun Su ◽  
Jesse Parker ◽  
Kazem Kazerounian ◽  
Horea Ilies
Keyword(s):  
Potential Energy ◽  
Molecular Dynamic Simulation ◽  
Molecular Dynamic ◽  
Dynamic Simulation ◽  
Multibody Dynamics ◽  
Energy State ◽  
Protein Structures ◽  
Rigid Bodies ◽  
Static Compliance ◽  
Protein Molecules

This paper presents an initial comparison of two approaches to energy minimization of protein molecules, namely the Molecular Dynamic Simulation and the Kineto-Static Compliance Method. Both methods are well established and are promising contenders to the seemingly insurmountable task of global optimization in the protein molecules potential energy terrain. The Molecular Dynamic Simulation takes the form of Constrained Multibody Dynamics of interconnected rigid bodies, as implemented at the Virtual Reality Application Center from Iowa State University. The Kineto-Static Compliance Method is implemented in the Protofold Computer package developed in the Mechanical Engineering Department at the University of Connecticut. The simulation results of both methods are compared through the trajectory of potential energy, the Root Mean Square Deviation (RMSD) of the alpha carbons, as well as based on visual observations. The preliminary results indicate that both techniques are very effective in converging the protein structure to a state with significantly less potential energy. At present, the converged solutions for the two methods are, however, different from each other and are very likely different from the global minimum potential energy state.

High-Fidelity Modeling of a Backhoe Digging Operation Using an Explicit Multibody Dynamics Code With Integrated Discrete Particle Modeling Capability

2013 ◽  
Author(s):  
Shahriar G. Ahmadi ◽  
Tamer M. Wasfy ◽  
Hatem M. Wasfy ◽  
Jeanne M. Peters
Keyword(s):  
Multibody Dynamics ◽  
Search Algorithm ◽  
Equations Of Motion ◽  
Friction Model ◽  
Rigid Bodies ◽  
Contact Friction ◽  
Contact Detection ◽  
High Fidelity ◽  
Normal Contact ◽  
Contact Search

A high-fidelity multibody dynamics model for simulating a backhoe digging operation is presented. The backhoe components including: frame, manipulator, track, wheels and sprockets are modeled as rigid bodies. The soil is modeled using cubic shaped particles for simulating sand with appropriate inter-particle normal and frictional forces. A penalty technique is used to impose both joint and normal contact constraints (including track-wheels, track-terrain, bucket-particles and particles-particles contact). An asperity-based friction model is used to model joint and contact friction. A Cartesian Eulerian grid contact search algorithm is used to allow fast contact detection between particles. A recursive bounding box contact search algorithm is used to allow fast contact detection between polygonal contact surfaces. The governing equations of motion are solved along with joint/constraint equations using a time-accurate explicit solution procedure. The model can help improve the performance of construction equipment by predicting the actuator and joint forces and the vehicle stability during digging for various vehicle design alternatives.

Modeling and simulation of planar flexible link manipulators with rigid tip connections to revolute joints

Robotica ◽  
2004 ◽  
Vol 22 (3) ◽  
pp. 285-300 ◽  
Author(s):  
S. M. Megahed ◽  
K. T. Hamza
Keyword(s):  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Link ◽  
Link Length ◽  
Finite Segment ◽  
Revolute Joints ◽  
Body Dynamics ◽  
Lumped Masses ◽  
Multi Body ◽  
Rigid Zones

This paper presents the basis of a mathematical model for simulation of planar flexible-link manipulators, taking into consideration the effect of higher stiffness zones at the link tips. The proposed formulation is a variation of the finite segment multi-body dynamics approach. The formulation employs a consistent mass matrix in order to provide better approximation than the traditional lumped masses often encountered in the finite segment approach. The formulation is implemented into a computational code and tested through three examples; cantilever beam, rotating beam and three-link manipulator. In these examples, the length of the rigid tips at both sides of each link ranges from 0% to 6.25% of the whole link length. The zones of higher stiffness at the link tips are treated as short rigid zones. The effect of the rigid zones is averaged along with some portions of the flexible links, thereby allowing further simplification of the dynamic equations of motion. The simulation results demonstrate the effectiveness of the proposed modeling technique and show the importance of not ignoring the effect of the rigid tips.

A Multibody System Model for Meshing Gears

2010 ◽  
Vol 44-47 ◽  
pp. 1273-1278 ◽  
Author(s):  
Liu Lei
Keyword(s):  
Multibody Dynamics ◽  
Multibody System ◽  
Gear Wheel ◽  
Numerical Approach ◽  
Computational Effort ◽  
Rigid Bodies ◽  
The Body ◽  
System Model ◽  
New Approach ◽  
Elastic Elements

As a type of numerical approach to dynamics of gears, multibody dynamics method can handle realistic cases of contact modeling with acceptable accuracy and considerably less computational effort. The ability to simulate contact between teeth has become an essential topic in multibody dynamics. Fully rigid method is not suited for a high quality of the analysis to take into account some elasticity in the model of meshing gear wheels. In our new approach the circumferentially rotatable rigid teeth and elastic elements composed of rotational spring-damper combinations are hereby put forward. The teeth and the body of each gear wheel are still regarded as rigid bodies, but they are connected with each other by elastic elements. Besides, Lankarani & Nikravesh Contact Model is utilized, which counts energy dissipation by means of viscous damping. Both large motions with revolutions and important elasticity are considered in this teeth-wheel multibody system model. Two examples are provided in which the simulation results of completely rigid method, the approach in [10], our new approach and finite element methods are compared. Comparisons indicate that our newly developed approach is more suitable for modeling multibody geared systems.

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