Theory of two-dimensional transformations

This paper proposes a new “heterogeneous” two-dimensional (2D) transformation group ⟨T, o⟩ to solve motion analysis/planning problems in robotics. In this theory, we use a 3×1 matrix to represent a transformation as opposed to a 3×3 matrix in the homogeneous formulation. First, this theory is as capable as the homogeneous theory, Because of the minimal size, its implementation requires less memory space and less computation time, and it does not have the rotational matrix inconsistency problem. Furthermore, the raw rotation angle θ is more useful than the trigonometric values, cos θ and sin θ, in the homogeneous transformations. This paper also discusses how to apply the group ⟨T, o⟩ to solve problems related to motion analysis/planning, trajectory generation, and others. This heterogeneous formulation has been successfully implemented in the MML software system for the autonomous mobile robot Yamabico-11 developed at the Naval Postgraduate School