Exact Displacement Analysis of Four-Link Spatial Mechanisms by the Direction Cosine Matrix Method

1984 ◽  
Vol 51 (4) ◽  
pp. 921-928 ◽  
Author(s):  
T. C. Huang ◽  
Y. Youm
Keyword(s):  
Local Coordinate System ◽  
Direction Cosine ◽  
Displacement Analysis ◽  
Spatial Mechanism ◽  
Spatial Mechanisms ◽  
Local Coordinates ◽  
Development Direction ◽  
Direction Cosine Matrix ◽  
Direction Cosines ◽  
Unit Vectors

A method of displacement analysis of the four-link spatial mechanism is developed. The results through this analysis will be exact solutions that can be obtained without resorting to numerical or iteration schemes. In the analysis, the position of a link in a mechanism can be fully defined if its direction and length are known. Therefore, this analysis involves the calculation of the unknown direction cosines and length of each link for a given configuration of the mechanism. In finding the direction cosines of the unknown unit vectors involved for each link and rotating axis, two types of coordinates, the global and the local, are generally used. Then, a direction cosine matrix between each local coordinate system and the global coordinates is established. Thus, the unknown direction cosines of the local coordinates, the links, and the rotating axes are obtained in global coordinates. In this development, direction cosine matrices are used throughout the analysis. As an illustration, the application of this method to the study of four-link spatial mechanisms, RGGR, RGCR, RRGG, and RRGC will be presented.

On the Displacement Analysis of Open-Loop Systems by the Direction Cosine Matrix Method

1987 ◽  
Author(s):  
Y. Youm ◽  
T. Yih
Keyword(s):  
Matrix Method ◽  
Direction Cosine ◽  
Open Loop ◽  
Open Loop System ◽  
Displacement Analysis ◽  
Transformation Matrices ◽  
Joint Axis ◽  
Cosine Transformation ◽  
Direction Cosine Matrix ◽  
Unit Vectors

Abstract In this paper, displacement analysis of a general spatial open-loop system and a computer algorithm for the workspace of the system are developed by applying the direction cosine matrix method. In using this method, one global coordinate system and two joint local coordinate systems must be predefined in order to formulate the direction cosine transformation matrices of the unit vectors of each joint axis and link vector. The 3 × 3 direction cosine transformation matrices for each joint axis and link vector are established based on the known geometric configurations, the preceding unit vectors, and the cofactor property of the direction cosine matrix. The use of cofactor property will provide a unique solution for the transformation matrix. A computer algorithm is developed to illustrate the workspace of spatial n-R open-loop systems projected onto the coordinate X-Y, Y-Z, and X-Z planes. Numerical examples are demonstrated for an industrial robot, an application to human upper extremity, and a hypothetical 9-link open-loop system.

Analysis of Spatial Open-Loop System by Means of Direction Cosine Transformation Matrices

1989 ◽  
Vol 111 (4) ◽  
pp. 508-512 ◽  
Author(s):  
T. C. Yih ◽  
Y. Youm
Keyword(s):  
Industrial Robot ◽  
Direction Cosine ◽  
Open Loop ◽  
Coordinate Systems ◽  
Open Loop System ◽  
Transformation Matrices ◽  
Joint Axis ◽  
Cosine Transformation ◽  
Direction Cosine Matrix ◽  
Unit Vectors

In this paper, an analytical approach for the displacement analysis of spatial openloop systems by means of direction cosine transformation matrices is presented. Two local coordinate systems at each joint are designated to formulate the direction cosine matrices, in recursive form, of the joint axis and link vector. Elements of the 3×3 direction cosine transformation matrices are computed based on the geometry of successive link elements, the unit vectors of preceding joint axis and link vector, and the cofactors of direction cosine matrix. The analysis using direction cosine matrix method will provide the “exact” joint positions in space. A computer algorithm is developed to investigate the workspaces of spatial n-R open-loop systems that projected onto the X-Y, Y-Z, and Z-X coordinate planes, respectively. Numerical examples for the workspaces of an industrial robot and the human upper extremity are illustrated.

A Dual Iterative Method for Displacement Analysis of Spatial Mechanisms

1994 ◽  
Author(s):  
Sean Thompson ◽  
Harry H. Cheng
Keyword(s):  
Iterative Method ◽  
Programming Language ◽  
Displacement Analysis ◽  
Spatial Mechanism ◽  
Dual Transformation ◽  
Transformation Matrices ◽  
Spatial Mechanisms

Abstract A dual iterative method for displacement analysis of a spatial mechanism is presented in this paper. The algorithm and formulation based upon 3 × 3 dual transformation matrices are succinct. They can be easily implemented in the CH programming language. The algorithm has been numerically verified by dual iterative displacement analysis of an RCCC four-link spatial mechanism.

Computer-Aided Displacement Analysis of Spatial Mechanisms

1994 ◽  
Author(s):  
Sean Thompson ◽  
Harry H. Cheng
Keyword(s):  
Programming Languages ◽  
Design Automation ◽  
Data Type ◽  
Complex Numbers ◽  
Dual Numbers ◽  
Displacement Analysis ◽  
Spatial Mechanism ◽  
Programming Paradigm ◽  
Spatial Mechanisms ◽  
Programming Techniques

Abstract Recently, Cheng (1993) introduced the CH programming language. CH is designed to be a superset of ANSI C with all programming features of FORTRAN. Many programming features in CH are specifically designed and implemented for design automation. Handling dual number as a basic built-in data type in the language is one example. Formulas with dual numbers can be translated into CH programming statements as easily as formulas with real and complex numbers. In this paper we will show that both formulation and programming with dual numbers are remarkably simple for analysis of complicated spatial mechanisms within the programming paradigm of CH. With computational capabilities for dual formulas in mind, formulas for analysis of spatial mechanisms are derived differently from those intended for implementation in computer programming languages without dual data type. We will demonstrate some formulation and programming techniques in the programming paradigm of CH through a displacement analysis of the RCRCR five-link spatial mechanism. A CH program that can obtain both numerical and graphical results for complete displacement analysis of the RCRCR mechanism will be presented.

On the shaft locations of two contra-rotating counterweights for balancing spatial mechanisms

1997 ◽  
Vol 211 (8) ◽  
pp. 567-578 ◽  
Author(s):  
S-T Chiou ◽  
J-C Tzou
Keyword(s):  
Root Mean Square ◽  
Mean Square ◽  
Spatial Mechanism ◽  
Spatial Mechanisms ◽  
Crank Mechanism ◽  
Frequency Term ◽  
Shaking Moment

It has been shown in a previous work that a frequency term of the shaking force of spatial mechanisms, whose hodograph is proved to be an ellipse, can be eliminated by a pair of contrarotating counterweights. In this work, it is found that the relevant frequency term of the shaking moment is minimized if the balancing shafts are coaxial at the centre of a family of ellipsoids, called isomomental ellipsoids, with respect to (w.r.t.) any point on an ellipsoid, as is also the root mean square (r.m.s.) of the relevant frequency term of the shaking moment. It can also be minimized even though the location of either shaft, but not both, is chosen arbitrarily on a plane. The location of the second shaft is then determinate. In order to locate the centre, a derivation for the theory of isomomental ellipsoids of a frequency term of the shaking moment of spatial mechanisms is given. It is shown that the r.m.s. of a frequency term shaking moment of a spatial mechanism w.r.t. the concentric centre of the isomomental ellipsoids is the minimum. Examples of a seven-link 7-R spatial linkage and a spatial slider-crank mechanism are included.

Attitude reconstruction from strap-down rate gyros using power series

2021 ◽  
pp. 1-19
Author(s):  
Habib Ghanbarpourasl
Keyword(s):  
Power Series ◽  
Taylor Series ◽  
Direction Cosine ◽  
Analytical Description ◽  
Double Precision ◽  
Angular Velocity Vector ◽  
Higher Order Terms ◽  
Direction Cosine Matrix ◽  
The Stability ◽  
Better Than

Abstract This paper introduces a power series based method for attitude reconstruction from triad orthogonal strap-down gyros. The method is implemented and validated using quaternions and direction cosine matrix in single and double precision implementation forms. It is supposed that data from gyros are sampled with high frequency and a fitted polynomial is used for an analytical description of the angular velocity vector. The method is compared with the well-known Taylor series approach, and the stability of the coefficients’ norm in higher-order terms for both methods is analysed. It is shown that the norm of quaternions’ derivatives in the Taylor series is bigger than the equivalent terms coefficients in the power series. In the proposed method, more terms can be used in the power series before the saturation of the coefficients and the error of the proposed method is less than that for other methods. The numerical results show that the application of the proposed method with quaternions performs better than other methods. The method is robust with respect to the noise of the sensors and has a low computational load compared with other methods.

Kinematic Analysis of Spatial Mechanisms by Means of Screw Coordinates. Part 2—Analysis of Spatial Mechanisms

1971 ◽  
Vol 93 (1) ◽  
pp. 67-73 ◽  
Author(s):  
M. S. C. Yuan ◽  
F. Freudenstein ◽  
L. S. Woo
Keyword(s):  
Computer Program ◽  
Kinematic Analysis ◽  
Static Force ◽  
Single Degree Of Freedom ◽  
Numerical Examples ◽  
Spatial Mechanism ◽  
Single Degree ◽  
Spatial Mechanisms ◽  
Torque Analysis ◽  
Basic Concepts

The basic concepts of screw coordinates described in Part I are applied to the numerical kinematic analysis of spatial mechanisms. The techniques are illustrated with reference to the displacement, velocity, and static-force-and-torque analysis of a general, single-degree-of-freedom spatial mechanism: a seven-link mechanism with screw pairs (H)7. By specialization the associated computer program is capable of analyzing many other single-loop spatial mechanisms. Numerical examples illustrate the results.

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