Constraint Forces and the Method of Auxiliary Generalized Speeds

2003 ◽  
Vol 70 (4) ◽  
pp. 568-574 ◽  
Author(s):  
S. Djerassi ◽  
H. Bamberger
Keyword(s):  
Multibody Systems ◽  
Computational Cost ◽  
Matrix Inversion ◽  
Constraint Forces ◽  
Constraint Force ◽  
Motion Equations ◽  
Reaction Forces

This paper deals with noncontributing forces, usually called constraint forces or reaction forces, arising in simple, nonholonomic multibody systems. These forces are related to two kinds of constraints, namely, kinematical constraints—derived from kinematical requirements, and auxiliary constraints, introduced for the purpose of constraint forces determination. Here, the method of “auxiliary generalized speeds” is used to bring into evidence constraint forces related to the two kinds of constraints. It is shown that auxiliary generalized speeds can always be chosen in a way that gives rise to additional equations each having one measure number of a constraint force as an unknown. Motion equations can thus be generated and solved without regard to constraint forces determination; and constraint forces can be determined with no matrix inversion, at a minimal computational cost.

Revisiting screw theory-based approaches in the constraint wrench analysis of robotic systems

Robotica ◽  
2021 ◽  
pp. 1-25
Author(s):  
Ehsan Sharafian M ◽  
Afshin Taghvaeipour ◽  
Maryam Ghassabzadeh S
Keyword(s):  
Multibody Systems ◽  
Screw Theory ◽  
Parallel Robot ◽  
Coordinate Frame ◽  
Force Transmission ◽  
Constraint Forces ◽  
Constraint Force ◽  
Inverse Dynamic ◽  
Transformation Matrices ◽  
Constraint Transformation

Abstract This paper aims at shedding lights on two approaches that were recently proposed for the constraint wrench analysis of robotic manipulators. Both approaches benefit from the Newton–Euler equations, screw notations, and constraint transformation matrices (CTM) to cope with the inverse dynamic problem of multibody systems. In the first approach, which is called the joint-based method, the constraint transformation matrices are derived directly from the kinematic constraints which are imposed on the rigid links by kinematic pairs. In the second approach, which is referred to as the link-based method; however, the constraint matrices are obtained based on the wrench transfer formula of each rigid link. In this study, by resorting to the definition of reciprocal screws, the former methodology is further enhanced to a new version as well. Moreover, based on the proposed modified joint-based CTM, constraint forces and moments distribution indices are introduced. The three constraint wrench analysis methodologies, two joint-based and one link-based, result in different CTMs and set of equations as well, which will be discussed in detail. In the end, on two case studies, a spherical four-bar linkage and a Delta parallel robot, the pros and cons of all three constraint wrench analysis methodologies are discussed, and the proposed indices will be examined. The numerical results reveal that, although all three methods identically compute the magnitude of the applied and constraint force and moment vectors, the joint-based approaches do not report the constraint components with respect to a specific coordinate frame. Moreover, it is shown that the proposed indices can approximately predict the constraint forces and moments distribution at joints, which can be used as force transmission indicators in multibody systems.

Effect of guide rail profile errors on the motion accuracy for a heavy-duty hydrostatic turntable

2019 ◽  
Vol 233 (15) ◽  
pp. 5350-5362 ◽  
Author(s):  
Yongsheng Zhao ◽  
Hongchao Wu ◽  
Congbin Yang ◽  
Ligang Cai ◽  
Zhifeng Liu
Keyword(s):  
Machine Tool ◽  
Reynolds Equation ◽  
Machining Accuracy ◽  
Guide Rail ◽  
Heavy Duty ◽  
Motion Equations ◽  
Rotating Speed ◽  
Proposed Model ◽  
Reaction Forces ◽  
Profile Errors

The motion accuracy of hydrostatic turntable is the key in improving the machining accuracy of heavy-duty machine tool. However, the motion accuracy of hydrostatic turntable depends not only on the offset load but also on the rotating speed of the turntable as well as the profile errors of the guide rails. In this paper, a simulation model is developed to analyze the effect of guide rail profile errors on the motion accuracy of hydrostatic turntable. The reaction forces of preload thrust bearing and hydrostatic circular oil pads are obtained based on the Reynolds equation of the lubricant film. The motion equations of hydrostatic turntable are derived in which the profile errors of two guide rails are considered. The results show that the motion accuracy of hydrostatic turntable can be affected by wavelength, amplitude of profile errors and speed, and offset load of turntable. Finally, the motion accuracy of heavy-duty hydrostatic turntable used in XCKA28105 type turning and milling composite machine tool is obtained by using the presented method. Comparing with the experimental results, the proposed model can be used to predict the machining accuracy caused by the profile errors of guide rails for any heavy-duty hydrostatic turntable.

On MBS Constraints and Projections

2021 ◽  
Author(s):  
Friedrich Pfeiffer
Keyword(s):  
Quadratic Form ◽  
Differential Equations ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Machine Process ◽  
Constraint Forces ◽  
Process Dynamics ◽  
Robot Tracking ◽  
Minimal Coordinates ◽  
Linear Differential

Abstract Constraints in multibody systems are usually treated by a Lagrange I - method resulting in equations of motion together with the constraint forces. Going from non-minimal coordinates to minimal ones opens the possibility to project the original equations directly to the minimal ones, thus eliminating the constraint forces. The necessary procedure is described, a general example of combined machine-process dynamics discussed and a specific example given. For a n-link robot tracking a path the equations of motion are projected onto this path resulting in quadratic form linear differential equations. They define the space of allowed motion, which is generated by a polygon-system.

Utilizing the Effort/Motion Approach in the Simulation of Interconnected Rigid Body Systems

1997 ◽  
Author(s):  
N. Duke Perreira
Keyword(s):  
Numerical Simulation ◽  
Rigid Body ◽  
Multibody Systems ◽  
Invariant Form ◽  
Second Law ◽  
Orthogonal Projections ◽  
Gauge Invariant ◽  
Motion Equations ◽  
Newton's Second Law ◽  
Constraint Surface

Abstract The effort/motion approach has been developed for use in designing, simulating and controlling multibody systems. Some aspects of each of these topics are discussed here. In the effort/motion formulation two sets of equations based on the orthogonal projections of a dimensional gauge invariant form of Newton’s Second Law occur. The projections are onto the normal and tangent directions of a dimensional gauge invariant constraint surface. The paper shows how these equations are obtained for a particular linkage with redundant effort and motion actuation. Two alternative Runga-Kutta based approaches for numerical simulation of the effort/motion equations are developed and applied in simulating the motion and determining the effort generated in the example linkage under various conditions. Oscillation about equilibrium positions, solutions with constant motion and with constant effort are given as examples of the approach.

A Vector Bond Graph Method of Kineto-Static Analysis for Spatial Multibody Systems

2013 ◽  
Vol 321-324 ◽  
pp. 1725-1729 ◽  
Author(s):  
Zhong Shuang Wang ◽  
Yang Yang Tao ◽  
Quan Yi Wen
Keyword(s):  
Static Analysis ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Multibody System ◽  
Graph Model ◽  
Bond Graph ◽  
Constraint Forces ◽  
Decoupling Method ◽  
Complete Form ◽  
Three Degrees Of Freedom

In order to increase the reliability and efficiency of the kineto-static analysis of complex multibody systems, the corresponding vector bond graph procedure is proposed. By the kinematic constraint condition, spatial multibody systems can be modeled by vector bond graph. For the algebraic difficulties brought by differential causality in system automatic kineto-static analysis, the effective decoupling method is proposed, thus the differential causalities in system vector bond graph model can be eliminated. In the case of considering EJS, the unified formulae of driving moment and constraint forces at joints are derived based on vector bond graph, which are easily derived on a computer in a complete form and very suitable for spatial multibody systems. As a result, the automatic kineto-static analysis of spatial multibody system on a computer is realized, its validity is illustrated by the spatial multibody system with three degrees of freedom.

Dynamical Equations With Non-Ideal Constraints: New and Old Alternatives

2007 ◽  
Author(s):  
Arun K. Banerjee ◽  
Mark Lemak
Keyword(s):  
Equations Of Motion ◽  
Alternative Form ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Constraint Force ◽  
Kane’S Method ◽  
Kane's Method ◽  
Ellipsoidal Surface ◽  
Ideal Constraint ◽  
The Ideal

This paper deals with the motion of mechanical systems with non-ideal constraints, defined as constraints where the forces associated with the constraint do work. The first objective of the paper is to show that two newly published formulations of equations of motion of systems with such non-ideal constraints are unnecessarily complex for situations where the non-ideal constraint force does not depend on the ideal constraint force, because they introduce and then eliminate these non-working constraint forces. We point out that a method already exists for nonideal constraints, namely, Kane’s equations, which are simpler because, among other things, they are based on automatic elimination of non-working constraints. The examples considered in these recent publications are worked out with Kane’s method to show the applicability and simplicity of Kane’s method for non-ideal constraints. A second objective of the paper is to present an alternative form of equations for systems where the non-ideal constraint force depends on the ideal constraint force, as in the case of Coulomb friction. The formulation is shown to lend itself naturally to also analyzing impact dynamics. The method is applied to the dynamics of a slug moving against friction on a moving ellipsoidal surface. Such a crude model may simulate, in essence, propellant motion in a tank in zero-g, or during docking of a spacecraft.

A Finite Element Geometrically Nonlinear Dynamic Formulation of Flexible Multibody Systems Using a New Displacements Representation

1997 ◽  
Vol 119 (4) ◽  
pp. 573-581 ◽  
Author(s):  
J. Mayo ◽  
J. Domi´nguez
Keyword(s):  
Finite Element ◽  
Multibody Systems ◽  
Strain Energy ◽  
Degrees Of Freedom ◽  
Synthesis Method ◽  
Geometrically Nonlinear ◽  
Motion Equations ◽  
Constant Stiffness ◽  
Elastic Nonlinearity ◽  
External Forces

In previous work (Mayo, 1993), the authors developed two geometrically nonlinear formulations of beams inflexible multibody systems. One, like most related methods, includes geometric elastic nonlinearity in the motion equations via the stiffness terms (Mayo and Domi´nguez, 1995), but preserving terms, in the expression for the strain energy, of a higher-order than most available formulations. The other formulation relies on distinguishing the contribution of the foreshortening effect from that of strain in modelling the displacement of a point. While including exactly the same nonlinear terms in the expression for the strain energy, the stiffness terms in the motion equations generated by this formulation are exclusively limited to the constant stiffness matrix for the linear analysis because the terms arising from geometric elastic nonlinearity are moved from elastic forces to inertial, reactive and external forces, which are originally nonlinear. This formulation was reported in a previous paper (Mayo et al, 1995) and used in conjunction with the assumed-modes method. The aim of the present work is to implement this second formulation on the basis of the finite-element method. If, in addition, the component mode synthesis method is applied to reduce the number of degrees of freedom, the proposed formulation takes account of the effect of geometric elastic nonlinearity on the transverse displacements occurring during bending without the need to include any axial vibration modes. This makes the formulation particularly efficient in computational terms and numerically more stable than alternative geometrically nonlinear formulations based on lower-order terms.

Analysis of Constraint Forces in Multibody Systems Based on the Differential Null Space Matrix Method

2019 ◽  
Author(s):  
Keisuke Kamiya
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Null Space ◽  
Governing Equations ◽  
Constraint Forces ◽  
Numerical Examples ◽  
Space Matrix ◽  
Rigid Multibody Systems ◽  
Rigid Multibody

Abstract This paper treats a problem to determine constraint forces in rigid mutibody systems. One of the most often applied algorithms for determination of constraint forces is based on the use of recursive Newton-Euler formalism. Another algorithm often applied for determination of constraint forces is based on the use of Lagrange multipliers. This paper presents a new method to determine constraint forces in rigid multibody systems. First relative displacements which violate the constraints, called anti-constraint relative displacements, are introduced, and governing equations which involve the constraint forces explicitly are derived. In the derived equations, the constraint forces appear independently from the Lagrange multipliers. Then, a method is proposed to determine the constraint forces by eliminating the Lagrange multipliers based on the methods proposed in previous papers by the author. The method is extended to have ability to treat systems with redundant constraints. Finally, validity of the proposed method is confirmed via numerical examples.

Dynamic Modeling and Analysis of a 3 PRS Parallel Mechanism Using Constrained Robotic Approach

2010 ◽  
Author(s):  
Meng-Shiun Tsai ◽  
Wei-Hsiung Yuan
Keyword(s):  
Parallel Mechanism ◽  
Driving Forces ◽  
Robotic Approach ◽  
Constraint Forces ◽  
Modeling And Analysis ◽  
Constraint Equations ◽  
Reaction Forces ◽  
Internal Forces ◽  
Moving Platform ◽  
Novel Model

In this paper, a novel model is developed to analyze a 3-PRS parallel mechanism. The 3-PRS mechanism consists of a moving platform which is manipulated by three actuated fingers. Since the fingertips are constrained by the moving platform, three holonomic constraint equations are derived and the associated constraint forces are studied based on the theory of differential geometry. The developed model could be utilized to separate the dynamics of the fingers and that of the moving platform such that the coupling behaviors of the fingers can be investigated. It is found that the reaction forces applied at the fingertips of the parallel manipulator include not only the driving forces to the moving platform but also the constraint forces. Besides, the constraint forces are determined to be internal forces which would not generate a net force or torque to drive the moving platform. Simulations are performed to study the dynamic behavior of the reaction forces and it is found that the constraint forces have significant effect on the actuated fingers.

Efficient Parallel Computer Simulation of the Motion Behaviors of Closed-Loop Multibody Systems

2007 ◽  
Author(s):  
Shanzhong Duan ◽  
Andrew Ries
Keyword(s):  
Parallel Computing ◽  
Multibody Systems ◽  
Closed Loop ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Parallel Computer ◽  
Constraint Forces ◽  
Computing Systems ◽  
Closed Loops

This paper presents an efficient parallelizable algorithm for the computer-aided simulation and numerical analysis of motion behaviors of multibody systems with closed-loops. The method is based on cutting certain user-defined system interbody joints so that a system of independent multibody subchains is formed. These subchains interact with one another through associated unknown constraint forces fc at the cut joints. The increased parallelism is obtainable through cutting joints and the explicit determination of associated constraint forces combined with a sequential O(n) method. Consequently, the sequential O(n) procedure is carried out within each subchain to form and solve the equations of motion while parallel strategies are performed between the subchains to form and solve constraint equations concurrently. For multibody systems with closed-loops, joint separations play both a role of creation of parallelism for computing load distribution and a role of opening a closed-loop for use of the O(n) algorithm. Joint separation strategies provide the flexibility for use of the algorithm so that it can easily accommodate the available number of processors while maintaining high efficiency. The algorithm gives the best performance for the application scenarios for n>>1 and n>>m, where n and m are number of degree of freedom and number of constraints of a multibody system with closed-loops respectively. The algorithm can be applied to both distributed-memory parallel computing systems and shared-memory parallel computing systems.

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