Reconciling Distance Metric Methods for Rigid Body Displacements

2009 ◽  
Author(s):  
Anurag Purwar ◽  
Qiaode Jeffrey Ge
Keyword(s):  
Rigid Body ◽  
Polar Decomposition ◽  
Fast Method ◽  
Robot Calibration ◽  
Mechanism Synthesis ◽  
Orthogonal Matrices ◽  
Higher Dimensional ◽  
Motion Approximation ◽  
Value Decomposition ◽  
And Control

In the last twenty years, researchers have proposed a few different methods to establish a norm (or, metric) for both planar and spatial rigid body displacements. Desire to meaningfully quantify a displacement composed of rotation and translation stems from a requirement to ascertain “distance” between two given displacements in applications, such as motion approximation and interpolation, mechanism synthesis, collision avoidance, positioning, and robot calibration and control. In this paper, we show that the various seemingly different shape independent norm calculation methods based on approximating displacements with higher dimensional rotations via orthogonal matrices, or polar decomposition (PD) and singular value decomposition (SVD) can be reconciled and unified in the mathematically compact and elegant framework of biquaternions. In the process, we also propose an elegant and fast method for such norm calculations.

A Distance Metric for Finite Sets of Rigid-Body Displacements via the Polar Decomposition

2006 ◽  
Vol 129 (8) ◽  
pp. 883-886 ◽  
Author(s):  
Pierre M. Larochelle ◽  
Andrew P. Murray ◽  
Jorge Angeles
Keyword(s):  
Singular Value Decomposition ◽  
Rigid Body ◽  
Polar Decomposition ◽  
Research Question ◽  
Singular Value ◽  
Invariant Metric ◽  
Euclidean Group ◽  
Coordinate Frames ◽  
Open Research ◽  
Value Decomposition

An open research question is how to define a useful metric on the special Euclidean group SE(n) with respect to: (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances that is useful for the synthesis and analysis of mechanical systems. We discuss a technique for approximating elements of SE(n) with elements of the special orthogonal group SO(n+1). This technique is based on using the singular value decomposition (SVD) and the polar decompositions (PD) of the homogeneous transform representation of the elements of SE(n). The embedding of the elements of SE(n) into SO(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements. The bi-invariant metric on SO(n+1) is then used to measure the distance between any two displacements. The result is a left invariant PD based metric on SE(n).

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

Sensorless Modal Stabilization for Three-Link Robotic Arm With Elastic Wrist Drive Belt

2001 ◽  
Author(s):  
Martin Hosek
Keyword(s):  
Rigid Body ◽  
Robotic Manipulator ◽  
Vibration Damping ◽  
Rigid Body Dynamics ◽  
Limiting Factor ◽  
Direct Drive ◽  
Stability And Control ◽  
Body Dynamics ◽  
Improved Stability ◽  
And Control

Abstract A control system for a three-link direct-drive robotic manipulator with inherent structural flexibilities is presented. The structural flexibilities introduce undesirable vibration modes which may affect operation of the robot motion controller, resulting in destabilization of the closed-loop system. This represents a major limiting factor for implementation of a conventional controller designed solely for the rigid body dynamics of the robotic manipulator. The fundamental idea in the presented approach is to use a composite controller which consists of a trajectory-tracking section designed for the rigid-body dynamics and a vibration-damping compensator added for attenuation of the dominant flexible dynamics. The vibration damping compensator operates on estimated states of the dominant flexible dynamics obtained from a reduced-order state observer. A mechanism is implemented which allows the robotic manipulator to move through or hold in positions where the dominant flexible dynamics is unobservable and uncontrollable. Results of laboratory tests document that the presented approach leads to improved stability and control performance.

scholarly journals Split-and-Combine Singular Value Decomposition for Large-Scale Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jengnan Tzeng
Keyword(s):  
Singular Value Decomposition ◽  
Large Scale ◽  
Semantic Analysis ◽  
Computational Cost ◽  
Matrix Decomposition ◽  
Singular Value ◽  
Fast Method ◽  
The Matrix ◽  
Scale Matrix ◽  
Value Decomposition

The singular value decomposition (SVD) is a fundamental matrix decomposition in linear algebra. It is widely applied in many modern techniques, for example, high- dimensional data visualization, dimension reduction, data mining, latent semantic analysis, and so forth. Although the SVD plays an essential role in these fields, its apparent weakness is the order three computational cost. This order three computational cost makes many modern applications infeasible, especially when the scale of the data is huge and growing. Therefore, it is imperative to develop a fast SVD method in modern era. If the rank of matrix is much smaller than the matrix size, there are already some fast SVD approaches. In this paper, we focus on this case but with the additional condition that the data is considerably huge to be stored as a matrix form. We will demonstrate that this fast SVD result is sufficiently accurate, and most importantly it can be derived immediately. Using this fast method, many infeasible modern techniques based on the SVD will become viable.

Validation of Measured Dynamic Data Using Rigid Body Response

2012 ◽  
Vol 55 (1) ◽  
pp. 25-39
Author(s):  
David Smallwood
Keyword(s):  
Rigid Body ◽  
Low Frequency ◽  
Linear Combinations ◽  
Resonant Frequencies ◽  
Low Frequencies ◽  
Spectral Density Matrix ◽  
Rigid Body Modes ◽  
Linear Fit ◽  
Value Decomposition ◽  
Body Response

As multiple axis vibration testing has become more widespread, it has become increasingly important to ensure the instrumentation is accurately portrayed in the instrumentation table. However, errors do occur. The method used in this paper to help uncover these errors is based on the condition that at low frequencies (below any resonant frequencies of the object being studied) the response is essentially rigid body. The spectral density matrix (SDM) at a low frequency, of many more than six response measurements, is decomposed using singular value decomposition (SVD). Under the assumption of rigid body response, it is assumed that the first six singular vectors are linear combinations of the six rigid body modes. The best linear fit is then calculated for this fit. The measurements are then removed one at a time, and the reduction in the fit error is calculated. It is assumed that if the removal of a measurement reduces the error significantly, that measurement is likely in error.

Identification and Observability Measure of a Basis Set of Error Parameters in Robot Calibration

1989 ◽  
Vol 111 (4) ◽  
pp. 513-518 ◽  
Author(s):  
Chia-Hsiang Menq ◽  
Jin-Hwan Borm ◽  
Jim Z. Lai
Keyword(s):  
Basis Set ◽  
Robot Calibration ◽  
Position Errors ◽  
Estimation Of Error ◽  
Observability Index ◽  
Value Decomposition ◽  
Svd Method ◽  
Observability Measure ◽  
Robot Configurations ◽  
Error Parameter

This paper presents a method of identifying a basis set of error parameters in robot calibration using the Singular Value Decomposition (SVD) method. With the method, the error parameter space can be separated into two: observable subspace and unobservable one. As a result, for a defined position error model, one can determine the dimension of the observable subspace, which is vital to the estimation of error parameters. The second objective of this paper is to study, when unmodeled error exists, the implications of measurement configurations in robot calibration. For selecting measurement configurations in calibration, and index is defined to measure the observability of the error parameters with respect to a set of robot configurations. As the observability index increases, the attribution of the position errors to the parameters becomes dominant and the effects of the measurement and unmodeled errors become less significant; consequently better estimation of the parameter errors can be obtained.

Dynamics and control of capture of a floating rigid body by a spacecraft robotic arm

2014 ◽  
Vol 33 (3) ◽  
pp. 315-332 ◽  
Author(s):  
Xiao-Feng Liu ◽  
Hai-Quan Li ◽  
Yi-Jun Chen ◽  
Guo-Ping Cai ◽  
Xi Wang
Keyword(s):  
Rigid Body ◽  
Robotic Arm ◽  
Dynamics And Control ◽  
And Control
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