scholarly journals Collisions of Constrained Rigid Body Systems with Friction

1998 ◽  
Vol 5 (3) ◽  
pp. 141-151 ◽  
Author(s):  
Haijun Shen ◽  
Miles A. Townsend
Keyword(s):  
Rigid Body ◽  
Tangential Velocity ◽  
Rigid Bodies ◽  
Physical Analysis ◽  
Double Pendulum ◽  
Physical Context ◽  
Collision Problem ◽  
Articulated Systems ◽  
Conventional Methods ◽  
Velocity Changes

A new approach is developed for the general collision problem of two rigid body systems with constraints (e.g., articulated systems, such as massy linkages) in which the relative tangential velocity at the point of contact and the associated friction force can change direction during the collision. This is beyond the framework of conventional methods, which can give significant and very obvious errors for this problem, and both extends and consolidates recent work. A new parameterization and theory characterize if, when and how the relative tangential velocity changes direction during contact. Elastic and dissipative phenomena and different values for static and kinetic friction coefficients are included. The method is based on the explicitly physical analysis of events at the point of contact. Using this method, Example 1 resolves (and corrects) a paradox (in the literature) of the collision of a double pendulum with the ground. The method fundamentally subsumes other recent models and the collision of rigid bodies; it yields the same results as conventional methods when they would apply (Example 2). The new method reformulates and extends recent approaches in a completely physical context.

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 ON INERTIAL MOTION OF AN ABSOLUTELY RIGID BODY ON A THREE-DEGREE SUSPENSION WITH LINKS OF FINITE LENGTH

2020 ◽  
Vol 2 (2) ◽  
pp. 6-17
Author(s):  
L Akulenko ◽  
◽  
N Bolotnik ◽  
D Leshchenko ◽  
E Palii ◽  
...  
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Exact Analytical Solution ◽  
Rigid Bodies ◽  
The Body ◽  
Theoretical Research ◽  
Double Pendulum ◽  
Two Degrees Of Freedom ◽  
Practical Applications ◽  
Wide Range

Papers on the dynamics of an absolutely rigid body with a fixed point generally assume that the mechanical system has three degrees of freedom. This is the situation when the body is attached to a fixed base by a ball-and-socket joint. On engineering systems one often encounters rigid bodies attached to a base by a two-degrees-of-freedom joint, consisting of a fixed axis and a movable one, which are mutually perpendicular. Such systems have two degrees of freedom, but the set of kinematically possible motions is quite rich. Dynamic analysis of the motion of a rigid body with a two-degree hinge in a force field is an integral part of the description of the action of mechanical actions of robotic systems. In recent decades, an increasingly closed role in the dynamics of rigid body systems has been played by manipulation robots consisting of a sequential chain of rigid links and controlled by means of torque drives in articulated joints. The same class of objects can be attributed to many biological systems that imitate, for example, the movements of a person or animal (walking, running, jumping). Two-link systems have a variety of practical applications and an almost equally wide range of areas of theoretical research. We note, in particular, the analysis of free and forced plane-parallel motion of a bundle of two rigid bodies connected by an ideal cylindrical hinge and simulating a composite satellite in outer space, a two-link manipulator, and an element of a crushing machine. The dynamic behavior of a rigid body in the gimbal suspension is a system, which can be interpreted as two-degree manipulator and used an element of more complex robotic structures. The linear mathematical model of two-link manipulator free oscillations with viscous friction in both its joints is a system, which reduces to the calculation scheme of double pendulum and allows the construction of exact analytical solution in the partial case. According to the research methodology, the proposed paper is close to works, where the motion by inertia of a plane two–rigid body hinged system was studied and devoted to the study of the motion of an absolutely rigid body on a power-to-power joint.

scholarly journals ClusterSLAM: A SLAM backend for simultaneous rigid body clustering and motion estimation

2021 ◽  
Author(s):  
Jiahui Huang ◽  
Sheng Yang ◽  
Zishuo Zhao ◽  
Yu-Kun Lai ◽  
Shi-Min Hu
Keyword(s):  
Motion Estimation ◽  
Rigid Body ◽  
Iterative Scheme ◽  
Dynamic Environments ◽  
Rigid Bodies ◽  
Factor Graph ◽  
Agglomerative Clustering ◽  
Multiple Objects ◽  
Graph Optimization ◽  
Dynamic Objects

AbstractWe present a practical backend for stereo visual SLAM which can simultaneously discover individual rigid bodies and compute their motions in dynamic environments. While recent factor graph based state optimization algorithms have shown their ability to robustly solve SLAM problems by treating dynamic objects as outliers, their dynamic motions are rarely considered. In this paper, we exploit the consensus of 3D motions for landmarks extracted from the same rigid body for clustering, and to identify static and dynamic objects in a unified manner. Specifically, our algorithm builds a noise-aware motion affinity matrix from landmarks, and uses agglomerative clustering to distinguish rigid bodies. Using decoupled factor graph optimization to revise their shapes and trajectories, we obtain an iterative scheme to update both cluster assignments and motion estimation reciprocally. Evaluations on both synthetic scenes and KITTI demonstrate the capability of our approach, and further experiments considering online efficiency also show the effectiveness of our method for simultaneously tracking ego-motion and multiple objects.

Unraveling Paradoxical Theories for Rigid Body Collisions

1991 ◽  
Vol 58 (4) ◽  
pp. 1049-1055 ◽  
Author(s):  
W. J. Stronge
Keyword(s):  
Frictional Force ◽  
Contact Point ◽  
Tangential Velocity ◽  
Velocity Slip ◽  
Rigid Bodies ◽  
Contact Points ◽  
Before And After ◽  
Limited Applicability ◽  
Work Done ◽  
Impulsive Reaction

A collision between two rigid bodies has a normal impulsive reaction at the contact point (CP). If the bodies are slightly rough and the contact points have a relative tangential velocity (slip), there is also a frictional force that opposes slip. Small initial slip can halt before contact terminates; when slip halts the frictional force changes and the collision process is separated into periods before and after halting. An energetically consistent theory for collisions with slip that halts is based on the work done by normal (nonfrictional) forces during restitution and compression phases. This theory clearly separates dissipation due to frictional forces from that due to internal irreversible deformation. With this theory, both normal and tangential components of the impulsive reaction always dissipate energy during collisions. In contrast, Newton’s impact law results in calculations of paradoxical increases in energy for collisions where slip reverses. This law relates normal components of relative velocity for the CP at separation and incidence by a constant (the coefficient of restitution e). Newton’s impact law is a kinematic definition for e that generally depends on the slip process and friction; consequently it has limited applicability.

The Use of Projective Geometry on the Mechanism System Motion and Stability of the Special Problems in Judgement

2012 ◽  
Vol 482-484 ◽  
pp. 1041-1044
Author(s):  
Xiao Zhuang Song ◽  
Ming Liang Lu ◽  
Tao Qin
Keyword(s):  
Rigid Body ◽  
Projective Geometry ◽  
Euclidean Geometry ◽  
Specific Problem ◽  
Zero Point ◽  
Rigid Bodies ◽  
Parallel Connection ◽  
Fixed Plane ◽  
Instantaneous Center ◽  
Proof Of Principle

In a principle of kinematics, when a rigid body is motion in a plane, and the fixed plane only the presence of a speed zero point -- the instantaneous center of velocity. In the mechanism of two rigid bodies be connected by two parallel connection links, why can the continuous relative translation? Where is the instantaneous center of velocity? ... ... The traditional Euclidean geometry theory can’t explain these phenomenon, must use projective geometry theory to solve. The actual motion of the mechanism is disproof in Euclidean geometry principle limitation. This paper introduces the required in projective geometry basic proof of principle, and applied to a specific problem.

On the Motion of Lineal Bodies Subject to Linear Damping

1997 ◽  
Vol 64 (1) ◽  
pp. 227-229 ◽  
Author(s):  
M. F. Beatty
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Rigid Body ◽  
General Class ◽  
Plane Motion ◽  
Rigid Bodies ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Linear Damping

Wilms (1995) has considered the plane motion of three lineal rigid bodies subject to linear damping over their length. He shows that these plane single-degree-of-freedom systems are governed by precisely the same fundamental differential equation. It is not unusual that several dynamical systems may be formally characterized by the same differential equation, but the universal differential equation for these systems is unusual because it is exactly the same equation for the three very different systems. It is shown here that these problems belong to a more general class of problems for which the differential equation is exactly the same for every lineal rigid body regardless of its geometry.

Closed-Form Determination of the Location of a Rigid Body by Seven In-Parallel Linear Transducers

1996 ◽  
Author(s):  
Carlo Innocenti
Keyword(s):  
Rigid Body ◽  
Linear Equations ◽  
Parallel Manipulators ◽  
Rigid Bodies ◽  
Body Location ◽  
Position And Orientation ◽  
General Geometry ◽  
Two Bodies

Abstract The paper presents an original analytic procedure for unambiguously determining the relative position and orientation (location) of two rigid bodies based on the readings from seven linear transducers. Each transducer connects two points arbitrarily chosen on the two bodies. The sought-for rigid-body location simply results by solving linear equations. The proposed procedure is suitable for implementation in control of fully-parallel manipulators with general geometry. A numerical example shows application of the reported results to a case study.

Numerical Simulation of Multibody Systems With Time Dependant Structure

1993 ◽  
Author(s):  
Pierre Joli ◽  
Madeleine Pascal ◽  
René Gibert
Keyword(s):  
Numerical Simulation ◽  
Multibody Systems ◽  
Computational Algorithm ◽  
Rigid Bodies ◽  
Three Dimensions ◽  
Integration Step ◽  
General Systems ◽  
Articulated Systems ◽  
Jump Velocity ◽  
Assembly Tasks

Abstract Current dynamic simulation programs are able to calculate the continuous motions of articulated systems or more general systems of rigid bodies in the absence of contact between members of the system or between the system and its environment. Some are able to simulate the effects of isolated contacts and impacts but none are able to simulate the motion with unrestricted multiple concurrent contacts. However, in special robotic programs such as robots performing assembly tasks or walking, it would be very interesting to simulate appropriate commands before implementing them on the robots. This paper develops intrinsic problems of collision to produce an efficient computational algorithm. This algorithm handles the detection of collision in three dimensions, the reduction of the integration step in order to avoid interpenetration between the bodies before impact, the jump velocity caused by a new collision and indicator magnitudes which determine the addition or deletion of constraints.

THE CALIBRATION OF AN ARRAY OF ACCELEROMETERS

2011 ◽  
Vol 35 (2) ◽  
pp. 251-267 ◽  
Author(s):  
Dany Dubé ◽  
Philippe Cardou
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Least Squares ◽  
Calibration Method ◽  
Calibration Procedure ◽  
Rigid Bodies ◽  
New Method ◽  
Scale Factors ◽  
Array Calibration ◽  
The One

An accelerometer-array calibration method is proposed in this paper by which we estimate not only the accelerometer offsets and scale factors, but also their sensitive directions and positions on a rigid body. These latter parameters are computed from the classical equations that describe the kinematics of rigid bodies, and by measuring the accelerometer-array displacements using a magnetic sensor. Unlike calibration schemes that were reported before, the one proposed here guarantees that the estimated accelerometer-array parameters are globally optimum in the least-squares sense. The calibration procedure is tested on OCTA, a rigid body equipped with six biaxial accelerometers. It is demonstrated that the new method significantly reduces the errors when computing the angular velocity of a rigid body from the accelerometer measurements.

On Higher Order Point-Line and the Associated Rigid Body Motions

2006 ◽  
Vol 129 (2) ◽  
pp. 166-172 ◽  
Author(s):  
Yi Zhang ◽  
Kwun-Lon Ting
Keyword(s):  
Rigid Body ◽  
Higher Order ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Dual Quaternion ◽  
Displacement Operator ◽  
Screw Motion ◽  
Rigid Body Motions ◽  
Point Line

This paper presents a study on the higher-order motion of point-lines embedded on rigid bodies. The mathematic treatment of the paper is based on dual quaternion algebra and differential geometry of line trajectories, which facilitate a concise and unified description of the material in this paper. Due to the unified treatment, the results are directly applicable to line motion as well. The transformation of a point-line between positions is expressed as a unit dual quaternion referred to as the point-line displacement operator depicting a pure translation along the point-line followed by a screw displacement about their common normal. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. A set of associated rigid body motions is obtained by applying an instantaneous rotation about the point-line. It shows that the ISA trihedrons of the associated rigid motions can be simply depicted with a set of ∞2 cylindroids. It also presents for a rigid body motion, the locus of lines and point-lines with common rotation or translation characteristics about the line axes. Lines embedded in a rigid body with uniform screw motion are presented. For a general rigid body motion, one may find lines generating up to the third order uniform screw motion about these lines.

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