New Approach to Determine 6DOF Position and Orientation of a Non-Orthogonal Coordinate System on the Object Using its Image

In this paper, a new method for determining position and orientation of a coordinate system using its image is presented. This coordinate system is a three dimensional non-orthogonal system in respect to the two dimensional and orthogonal camera coordinate system. In real world, it’s exactly easy to select three directions on an object so that they don’t be orthogonal and on a plane. The image of this non-orthogonal coordinate system on the camera image plane is a two dimensional coordinate system. This image is obtained by a nonlinear mapping between three dimensional worlds coordinate and two dimensional image coordinate. In this paper, we review geometric relationships between a direction of a vector on the object and its image that was presented in paper [18] months ago. Then, using these relationships for three arbitrary non-orthogonal directions which are not on a plane, a system of 15 nonlinear equations is established, and by solving it, nine unknowns are extracted. Because of the importance of the sign of these unknowns to determine true lengths and angels, it’s essential to run this system of nonlinear equations in eight cases and then best answer with right signs can be extracted. The results of this theory have been examined using simulation and programs.
In paper [18] we have to select three orthogonal vectors on an object. Since world is 3D, in some cases it is exactly difficult to choose all three directions with proper length and maybe we have to choose third vector (which is in depth) with a short length and it increase errors in finding position and orientation. But in this paper we don’t limit directions to be orthogonal, so all three directions can be in proper length and it decreases the errors.