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13
Solving the 6R inverse position problem using a genericcase solution methodology
 Mech. Mach. Theory
, 1991
"... AlmU'netThis paper considers the computation of all solutions to the inverse position problem for general sixrevolutejoint manipulators. Instead of reducing the problem to one highly complicated inputoutput equation, we work with a system of I I very simple polynomial equations. Although the to ..."
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Cited by 21 (1 self)
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AlmU'netThis paper considers the computation of all solutions to the inverse position problem for general sixrevolutejoint manipulators. Instead of reducing the problem to one highly complicated inputoutput equation, we work with a system of I I very simple polynomial equations. Although the total degree of the system is large (1024), using the "method of the generic case " we show numerically that the generic number of solutions is 16, in agreement with previous works. Moreover. we present an elEcient nun~rical method for finding all 16 solutions, based on coe~icientparameter polynomial continuation. We present a set of 41 test problems, on which the algorithm used an average of le ~ than l0 s of CPU time on an IBM 3703090 in double precision FORTRAN. The methodology applies equally well to other problems in kinematics that can be formulated as polynomial systems. I.
Algebraic Methods for Image Processing and Computer Vision
 IEEE Transactions on Image Processing
, 1996
"... Many important problems in image processing and computer vision can be formulated as the solution of a system of simultaneous polynomial equations. Crucial issues include the uniqueness of solution and the number of solutions (if not unique), and how to find numerically all the solutions. The goal o ..."
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Cited by 8 (2 self)
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Many important problems in image processing and computer vision can be formulated as the solution of a system of simultaneous polynomial equations. Crucial issues include the uniqueness of solution and the number of solutions (if not unique), and how to find numerically all the solutions. The goal of the present paper is to introduce to engineers and scientists some mathematical tools from algebraic geometry which are very useful in resolving these issues. Threedimensional motion/structure estimation is used as the context. However, these tools should also be helpful in other areas including surface intersection in CAD, and inverse position problems in kinematics/robotics. The tools to be described are: B'ezout numbers, Grobner bases, homotopy methods, and a powerful theorem which states that under rather general conditions one can draw general conclusions on the number of solutions of a polynomial system from a single numerical example. I. Introduction M ANY important problems in ...
Resultants and loop closure
 International Journal of Quantum Chemistry
, 2005
"... ABSTRACT: The problem of tripeptide loop closure is formulated in terms of the 3 angles {�i} i�1 describing the orientation of each peptide unit about the virtual axis joining the C � atoms. Imposing the constraint that at the junction of two such units the bond angle between the bonds C�ON and C�OC ..."
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Cited by 5 (2 self)
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ABSTRACT: The problem of tripeptide loop closure is formulated in terms of the 3 angles {�i} i�1 describing the orientation of each peptide unit about the virtual axis joining the C � atoms. Imposing the constraint that at the junction of two such units the bond angle between the bonds C�ON and C�OC is fixed at some prescribed value � results in a system of three bivariate polynomials in ui: � tan �i/2 of degree 2 in each variable. The system is analyzed for the existence of common solutions by making use of resultants, determinants of matrices composed of the coefficients of two (or more) polynomials, whose vanishing is a necessary and sufficient condition for the polynomials to have a common root. Two resultants are compared: the classical Sylvester resultant and the Dixon resultant. It is shown that when two of the variables are eliminated in favor of the third, a polynomial of degree 16 results. To each one of its real roots, there is a corresponding common zero of the system. To each such zero, there corresponds a consistent conformation of the chain. The Sylvester method can find these zeros among the eigenvalues of a 24 � 24 matrix. For the Dixon approach, after removing extraneous factors, an optimally sized eigenvalue problem of size 16 � 16 results. Finally, the easy extension to the more general problem of triaxial loop closure is presented and an algorithm for implementing the method on arbitrary chains is
Linear Algebra and Numerical Algorithms Using Dual Numbers
"... Dual number algebra is a powerful mathematical tool for the kinematic and dynamic analysis of spatial mechanisms. With the purpose of exploting new applications, in this paper are presented the dual version of some classical linear algebra algorithms. These algorithms have been tested for the positi ..."
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Cited by 2 (2 self)
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Dual number algebra is a powerful mathematical tool for the kinematic and dynamic analysis of spatial mechanisms. With the purpose of exploting new applications, in this paper are presented the dual version of some classical linear algebra algorithms. These algorithms have been tested for the position analysis of the RCCC mechanism and computational improvements over existing methods obtained.
Linear Dual Algebra Algorithms and their Application to Kinematics
"... Mathematical and mechanical entities such as line vectors, screws and wrenches can be conveniently represented within the framework of dual algebra. Despite the applications received by this type of algebra, less developed appear the numerical linear algebra algorithms within the field of dual numb ..."
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Cited by 1 (1 self)
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Mathematical and mechanical entities such as line vectors, screws and wrenches can be conveniently represented within the framework of dual algebra. Despite the applications received by this type of algebra, less developed appear the numerical linear algebra algorithms within the field of dual numbers. In this paper will be summarized different basic algorithms for handling vectors and matrices of dual numbers. It will be proposed an original application to finite and infinitesimal rigid body motion analysis.
On the Geometry of Spatial Polygons and Screw Polygons
 TRANSACTIONS OF THE ASME, JOURNAL OF MECHANICAL DESIGN
, 1996
"... In this paper we study spatial polygons which are closed figures composed of ordered sets of lines and their common normals. It is shown how to solve for up to six of the parameters which define a polygon's geometry, after all the other parameters are specified. It is also shown that when screws are ..."
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Cited by 1 (1 self)
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In this paper we study spatial polygons which are closed figures composed of ordered sets of lines and their common normals. It is shown how to solve for up to six of the parameters which define a polygon's geometry, after all the other parameters are specified. It is also shown that when screws are used instead of lines, the resulting figures, called screw polygons, can be analyzed in order to determine up to twelve parameters, provided all the other parameters which define the screws and the screw polygon are known. Finally, it is pointed out that these results have many potential uses in kinematics.
Position Analysis, via Velocity and Acceleration, in the Bennett Linkage
"... Introduction. Since its discovery, at the beginning of this century, the Bennett linkage has been the object of uncountable studies. Among the many exceptional properties of this linkage, the most remarkable is that it is an overconstrained linkage; i.e. eventhough the KutzbachGrubler criterion pr ..."
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Introduction. Since its discovery, at the beginning of this century, the Bennett linkage has been the object of uncountable studies. Among the many exceptional properties of this linkage, the most remarkable is that it is an overconstrained linkage; i.e. eventhough the KutzbachGrubler criterion predicts that the Bennett linkage is a statically undetermined structure, it is really a linkage of mobility one. Baker [1988,1993] has studied overconstrained linkages, and the Bennett linkage in particular, extensively. In these contributions, the reader can find a comprehensive bibliography on the subject. A closely related problem is that of determining finite or global mobility from local or infinitesimal conditions; i.e. if a linkage, in a given position, has a solvable velocity, acceleration, and higher order analyses, what can be said about the global mobility  the mobility over a finite interval of the input variables  of the linkage? The present contribution provides an a
Implementation Of Planar Hybris Active/passive ForceFeedback User Input Device
, 2002
"... x 1 ..."