3D Euler Angle Computation via Universal Matrix Pattern
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Solution Overview
Problem
Existing methods for computing 3D Euler angles are cumbersome and require manual identification of singular matrix elements, making it difficult to compute Euler angles in all sequences at once.
Innovation Solution
A method and apparatus for calculating relative orientations of rigid bodies in space by determining 3D coordinate systems and implementing a single set of Euler angle equations, irrespective of the rotation sequence or convention chosen, using 3×3 rotation matrices.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If manual identification of single matrix element is used to compute Euler angles, then computation can be performed for specific rotation sequences, but the method becomes cumbersome and requires different equation sets for each rotation sequence
Solution Approach 1:
The patent develops a universal algorithm that can compute Euler angles for any rotation sequence (XYZ, XZY, YXZ, YZX, ZXY, ZYX) using a single consistent set of equations. The method identifies patterns in the 3×3 rotation matrices that allow the same computational approach to work across all six possible rotation sequences, eliminating the need for separate equation sets for each convention.
Solution Approach 2:
The invention changes the approach from manually identifying specific matrix elements to using systematic pattern recognition in the rotation matrices. By analyzing the structure of 3×3 rotation matrices and identifying consistent mathematical relationships across different rotation sequences, the method transforms the computation into a parameter-based solution that adapts to any rotation convention through the same algorithmic framework.
2Adaptability or versatility
If different rotation sequences are used, then different 3×3 matrix element patterns emerge, but this requires multiple sets of equations to solve for Euler angles
Solution Approach 1:
The patent creates a single universal algorithm that handles all six rotation sequences (XYZ, XZY, YXZ, YZX, ZXY, ZYX) without requiring separate computation paths. By identifying invariant patterns in the rotation matrices, the method achieves universal applicability across all rotation conventions, reducing computational time by eliminating the need to select and implement different equation sets for different sequences.
3Measurement precision
If conventional methods are used to calculate Euler angles, then results are accurate for specific conventions, but the process requires manual identification of matrix elements making it difficult to compute all sequences at once
Solution Approach 1:
The patent implements an automated algorithm that self-identifies the appropriate matrix elements and computational paths based on the input rotation sequence. The system automatically analyzes the 3×3 rotation matrix structure and applies the correct mathematical relationships without requiring manual intervention to identify which elements to use, making the computation process easier while maintaining accuracy across all rotation sequences.
Data Source
AI summary
Systems, methods, apparatuses, and computer program products for computing three-dimensional (3D) Euler angles through a distinctive matrices pattern. A method for calculating relative orientations of rigid bodies in space may include determining a three-dimensional (3D) coordinate system of a rigid body D at a time T. The method may also include determining a 3D coordinate system of a rigid body E at time T. The method may further include determining a relative orientation at time T of the rigid body E in the 3D coordinate system of the rigid body D. In addition, the method may include calculating a final relative orientation of the rigid body E in the 3D coordinate system of the rigid body D by implementing a single set of Euler angle equations irrespective of a rotation sequence or a convention chosen.