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# Generation of the Trajectory of an Object Under Geometrical Contraints

IP.com Disclosure Number: IPCOM000120128D
Original Publication Date: 1991-Mar-01
Included in the Prior Art Database: 2005-Apr-02
Document File: 2 page(s) / 70K

IBM

Okano, A: AUTHOR

## Abstract

Disclosed is a method of generating the trajectory of an object when it moves from an initial position to a destination under geometrical constraints.

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Generation of the Trajectory of an Object Under Geometrical Contraints

Disclosed is a method of generating the trajectory of an
object when it moves from an initial position to a destination under
geometrical constraints.

The method consists of the following steps:  1) generation of
equations representing constraints among geometrical elements, (2)
calculation of the Groebner base of the simultaneous equations
obtained in step 1, (3) determination of independent variables, using
dependency relations among variables in the equations, (4)
calculation of the trajectories of the independent variables, and (5)
calculation of the trajectories of the dependent variables by
substituting the trajectories of the independent variables into the
Groebner base.

The trajectory of a moving object can be determined from
geometrical constraints among objects and the initial and final
points of the moving object.

The position of an object is represented by a 4 x 4 matrix as
follows:

(Image Omitted)

where N, O, A vectors denote an orientation and P vector
denotes a location.

There are twelve variables in the matrix. The trajectory of an
object is represented by the trajectories of the twelve variables.
There are six constraints among the variables, as follows:

(Image Omitted)

First, algebraic equations representing the given constraints
among geometrical elements are generated, where the geometric
elements are vertices, edges, and faces, and the constraints among
geometrical elements are distances and angle...