On the Metrics of Rigid Body Displacements for Infinite and Finite Bodies

1995 ◽  
Vol 117 (1) ◽  
pp. 41-47 ◽  
Author(s):  
J. M. R. Martinez ◽  
J. Duffy
Keyword(s):  
Rigid Body ◽  
Critical Analysis ◽  
Rigid Bodies ◽  
Euclidean Group ◽  
Finite Displacement ◽  
Kinematic Mapping ◽  
Planar Group

This paper presents a critical analysis of the metric of rigid body displacements obtained from the so-called kinematic mapping. It is shown that this metric is not suitable for finite rigid bodies. The paper also addresses the metrics obtained for the planar group, as it can be regarded as a subgroup of the group of all rigid body displacements, which is denoted here as the Euclidean group. Finally, the paper proposes some metrics for the set of spatial and planar displacements for a finite rigid body, undergoing a finite displacement.

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.

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.

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.

scholarly journals Formulation and Simulations of the Conserving Algorithm for Feedback Stabilization on Rigid Body Rotations

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Yong-Ren Pu ◽  
Thomas A. Posbergh
Keyword(s):  
Rigid Body ◽  
Closed Loop ◽  
Numerical Algorithms ◽  
Open Loop ◽  
Rigid Bodies ◽  
Feedback Stabilization ◽  
Conserved Quantities ◽  
Control Laws ◽  
Loop Control ◽  
Closed Loop Systems

The problem of stabilization of rigid bodies has received a great deal of attention for many years. People have developed a variety of feedback control laws to meet their design requirements and have formulated various but mostly open loop numerical algorithms for the dynamics of the corresponding closed loop systems. Since the conserved quantities such as energy, momentum, and symmetry play an important role in the dynamics, we investigate the conserved quantities for the closed loop control systems which formally or asymptotically stabilize rigid body rotation and modify the open loop numerical algorithms so that they preserve these important properties. Using several examples, the authors first use the open loop algorithm to simulate the tumbling rigid body actions and then use the resulting closed loop one to stabilize them.

Study on Vibration Mitigation of Connected Cabinets Storing Electronics Subjected to Seismic Input Using Elasto-Plastic Damper

2020 ◽  
Author(s):  
Atsuhiko Shintani ◽  
Takuma Yoshida ◽  
Chihiro Nakagawa ◽  
Tomohiro Ito
Keyword(s):  
Rigid Body ◽  
Seismic Wave ◽  
Power Plants ◽  
Rigid Bodies ◽  
Single System ◽  
Maximum Acceleration ◽  
Worst Case ◽  
Seismic Input ◽  
Entire Plant ◽  
Rocking Motion

Abstract This paper deals with the motion of coupled cabinets containing electronics subjected to seismic input. In power plants, chemical plants, etc., several rectangular cabinets containing important electronics are always lined up in the control center. These electronics are necessary for the control of the entire plant; thus, when they are damaged, the entire plant cannot be controlled, and a serious accident may occur. These cabinets are frequently put directly on the floor. Thus, it is perceived that in the worst case, cabinets may turn over by rocking motion during earthquakes and electronics may break. Moreover, even when the cabinets do not overturn, there is a concern about a large acceleration applied to the internal electronics due to the seismic waves. Hence, the need to develop methods that can reduce rocking motion and prevent electronics damage simultaneously. First, we consider the single cabinet with electronics. The cabinet is modeled as a rotating rigid body around its corner. The internal electronics are modeled as a rigid body moving in the translational direction in the cabinet. This system is referred to as single system. We input a seismic wave to the single system and investigate the rocking angle of the cabinet and the acceleration of the electronics in the cabinet. Consequently, we consider the adjacent cabinets connected by an elasto-plastic damper containing electronics. The cabinets are modeled as rotating rigid bodies. The internal electronics are modeled as rigid bodies moving in the translational direction in the cabinets. The whole system is known as a connected system. The elasto-plastic damper has bilinear hysteretic characteristics and can absorb the energy of earthquake inputs. We input the same seismic wave to the connected system to obtain the rocking angle of cabinets and the acceleration of electronics in the connected system. In these simulations, it is assumed that cabinets do not collide with each other. Then, we investigate the effect of the parameters of the elasto-plastic damper suppressing the rocking angle of the cabinets and the acceleration of electronics. Finally, we compare the maximum rocking angle and the maximum acceleration of the single system with that of the connected system and consider an ideal method to reduce the rocking angle and the acceleration simultaneously.

Contact-space resolution model for a physically consistent view of simultaneous collisions in articulated-body systems: theory and experimental results

2020 ◽  
Vol 39 (10-11) ◽  
pp. 1239-1258
Author(s):  
Shameek Ganguly ◽  
Oussama Khatib
Keyword(s):  
System Dynamics ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Null Space ◽  
Rigid Bodies ◽  
Computationally Efficient ◽  
Space Resolution ◽  
Resolution Model ◽  
Space Dynamics ◽  
Contact Space

Multi-surface interactions occur frequently in articulated-rigid-body systems such as robotic manipulators. Real-time prediction of contact-interaction forces is challenging for systems with many degrees of freedom (DOFs) because joint and contact constraints must be enforced simultaneously. While several contact models exist for systems of free rigid bodies, fewer models are available for articulated-body systems. In this paper, we extend the method of Ruspini and Khatib and develop the contact-space resolution (CSR) model by applying the operational space theory of robot manipulation. Through a proper choice of contact-space coordinates, the projected dynamics of the system in the contact space is obtained. We show that the projection into the dynamically consistent null space preserves linear and angular momentum in a subspace of the system dynamics complementary to the joint and contact constraints. Furthermore, we illustrate that a simultaneous collision event between two articulated bodies can be resolved as an equivalent simultaneous collision between two non-articulated rigid bodies through the projected contact-space dynamics. Solving this reduced-dimensional problem is computationally efficient, but determining its accuracy requires physical experimentation. To gain further insights into the theoretical model predictions, we devised an apparatus consisting of colliding 1-, 2-, and 3-DOF articulated bodies where joint motion is recorded with high precision. Results validate that the CSR model accurately predicts the post-collision system state. Moreover, for the first time, we show that the projection of system dynamics into the mutually complementary contact space and null space is a physically verifiable phenomenon in articulated-rigid-body systems.

Receptance Series for Systems Possessing “Rigid Body” Modes

1962 ◽  
Vol 66 (618) ◽  
pp. 394-397 ◽  
Author(s):  
G. M. L. Gladwell ◽  
R. E. D. Bishop ◽  
D. C. Johnson
Keyword(s):  
Rigid Body ◽  
Natural Frequencies ◽  
Series Representation ◽  
Forced Vibrations ◽  
Rigid Bodies ◽  
Points Of Interest ◽  
Elastic Systems ◽  
Rigid Body Modes ◽  
Simple Systems

SummaryCertain elastic systems may not only vibrate freely at proper (non-zero) natural frequencies, but may also move as rigid bodies. Such systems have “rigid body” modes which behave like principal modes corresponding to zero natural frequencies. These modes may be disregarded in the series representation of static distortions of such systems but must be taken into account in the representation of forced vibrations.This note is concerned with the series representation of receptances of certain simple systems of this type, namely strings, bars, shafts and beams. These systems were discussed in reference 1, but there the rigid body modes were omitted. As the matter appears to raise some points of interest, a discussion of it seems to be called for. A similar analysis to that presented here may be applied to other unsupported, or partially supported systems, such as an unsupported plate.

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