A Linear Algebra Approach to the Analysis of Rigid Body Displacement From Initial and Final Position Data

1982 ◽  
Vol 49 (1) ◽  
pp. 213-216 ◽  
Author(s):  
A. J. Laub ◽  
G. R. Shiflett
Keyword(s):  
Rigid Body ◽  
Linear Algebra ◽  
Initial Position ◽  
The Body ◽  
Final Position ◽  
Explicit Formulas ◽  
Algebra Approach ◽  
Other Information ◽  
Position Data ◽  
Rigid Body Displacement

The location and orientation of a rigid body in space can be defined in terms of three noncollinear points in the body. As the rigid body is moved through space, the motion may be described by a series of rotations and translations. The sequence of displacements may be conveniently represented in matrix form by a series of displacement matrices that describe the motion of the body between successive positions. If the rotations and translations (and hence the displacement matrix) are known then succeeding positions of a rigid body may be easily calculated in terms of the initial position. Conversely, if successive positions of three points in the rigid body are known, it is possible to calculate the parameters of the corresponding rotation and translation. In this paper, a new solution is presented which provides explicit formulas for the rotation and translation of a rigid body in terms of the initial and final positions of three points fixed in the rigid body. The rotation matrix is determined directly whereupon appropriate rotation angles and other information can subsequently be calculated if desired.

Comparison of Methods for Determining Screw Parameters of Finite Rigid Body Motion From Initial and Final Position Data

1989 ◽  
Author(s):  
R. G. Fenton ◽  
X. Shi
Keyword(s):  
Rigid Body ◽  
Rigid Body Motion ◽  
The Other ◽  
Body Motion ◽  
Final Position ◽  
Comparison Of Methods ◽  
Rodrigues Formula ◽  
Position Data ◽  
Data Error

Abstract Five methods for determining the screw parameters of finite rigid body motion using position data of three non-collinear points are compared on the basis of their efficiency, accuracy, and sensitivity to data error. It is found that the method based on Rodrigues’ Formula (Bottema & Roth’s method) is the most efficient. Angeles’ method and Laub & Shiflett’s method provide approximately the same level of accuracy, which is superior to that of the other methods. In terms of sensitivity, Bottema & Roth’s method is preferable On the basis of this study it is recommended that Bottema & Roth’s method to be used if uncertainty exists in the data since it can provide a solution efficiently, accurately, and it is the least sensitive to data error.

Comparison of Methods for Determining Screw Parameters of Finite Rigid Body Motion From Initial and Final Position Data

1990 ◽  
Vol 112 (4) ◽  
pp. 472-479 ◽  
Author(s):  
R. G. Fenton ◽  
Xiaolun Shi
Keyword(s):  
Rigid Body ◽  
Rigid Body Motion ◽  
The Other ◽  
Body Motion ◽  
Final Position ◽  
Comparison Of Methods ◽  
Rodrigues Formula ◽  
Position Data ◽  
Data Error

Five methods for determining screw parameters of finite rigid body motion, using position data of three noncollinear points, are compared on the basis of their efficiency, accuracy, and sensitivity to data error. It is found that the method based on Rodrigues’ Formula (Bottema and Roth’s method) is the most efficient. Angeles’ method and Laub and Shiflett’s method provide approximately the same level of accuracy, which is superior to that of the other methods. In terms of sensitivity, Bottema and Roth’s method is preferable. On the basis of this study it is recommended that Bottema and Roth’s method be used if uncertainty exists in the data, since it can provide a solution efficiently, accurately and it is the least sensitive to data error.

Completely Specified Displacements of a Rigid Body and Their Application in the Direct Kinematics of In-Parallel Mechanisms

1999 ◽  
Vol 121 (4) ◽  
pp. 485-491
Author(s):  
C. Mavroidis
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Algebraic Solution ◽  
Final Position ◽  
Parallel Mechanisms ◽  
Direct Kinematics

In this paper we study the type of completely specified displacements of a rigid body where the distances of six points on the body, between their final and initial positions, have been specified. A method is presented to calculate the screws associated with these displacements. It is shown that there are 24 different screws at the most that will bring the rigid body from the initial to its final position to satisfy the six distances. If the six points are co-planar or symmetrical with respect to a plane then the number of screws reduces to 16. The method to calculate the screws associated with these completely specified displacements, is used to obtain a simple algebraic solution to the direct kinematics problem of a special type of in-parallel mechanisms: the mechanisms where the moving and the fixed platforms are the same.

A Linear Algebra Approach to the Analysis of Rigid Body Velocity From Position and Velocity Data

1983 ◽  
Vol 105 (2) ◽  
pp. 92-95 ◽  
Author(s):  
A. J. Laub ◽  
G. R. Shiflett
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Matrix Theory ◽  
The Body ◽  
Instantaneous Velocity ◽  
Body Motion ◽  
Translational Velocity ◽  
Velocity Data ◽  
Algebra Approach ◽  
Body Velocity

The instantaneous velocity of a rigid body in space is characterized by an angular and translational velocity. By representing the angular velocity as a matrix and the translational component as a vector the velocity of any point in the rigid body may be found if the position of the point and the parameters of the angular and translational velocities are known. Alternatively, the parameters of the rigid body velocity may be determined if the velocity and position of three points fixed in the body are known. In this paper, a new matrix-theory-based method is derived for determining the instantaneous velocity parameters of rigid body motion in terms of the velocity and position of three noncollinear points fixed in the body. The method is shown to possess certain advantages over traditional vectoral solutions to the same problem.

Completely Specified Displacements of a Rigid Body and Their Application in the Direct Kinematics of In-Parallel Mechanisms

1998 ◽  
Author(s):  
Constantinos Mavroidis
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Algebraic Solution ◽  
Final Position ◽  
Parallel Mechanisms ◽  
Direct Kinematics

Abstract In this paper we study the type of completely specified displacements of a rigid body where the distances of six points on the body, between their final and initial positions, have been specified. A method is presented to calculate the screws associated with these displacements. It is shown that there are 24 different screws at the most that will bring the rigid body from the initial to its final position to satisfy the six distances. If the six points are co-planar or symmetrical with respect to a plane then the number of screws reduces to 16. The method to calculate the screws associated with these completely specified displacements, is used to obtain a simple algebraic solution to the direct kinematics problem of a special type of in-parallel mechanisms: the mechanisms where the moving and the fixed platforms are the same.

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