CONSTRUCTION OF CONFIDENCE REGIONS FOR UNCERTAIN SPATIAL DISPLACEMENTS WITH DUAL RODRIGUES PARAMETERS.

Area of Science:

• Robotics and Kinematics
• Computational Geometry
• Spatial Metrology

Background:

• Kinematic confidence regions are crucial for understanding uncertainty in spatial displacements.
• Existing methods like the Rotational and Translational Confidence Limit (RTCL) use Euler angles and translation parameters.
• There is a need for more robust and efficient methods to define and compute these regions.

Purpose of the Study:

• To develop a novel approach for computing kinematic confidence regions using dual quaternion algebra.
• To compare the efficacy of the proposed dual-Rodrigues parameter formulation against the established RTCL method.
• To demonstrate improved accuracy and compactness in swept volume computation.

Main Methods:

• Utilizing dual quaternion algebra to compute mean and relative spatial displacements.
• Employing dual Rodrigues parameters derived from dual quaternions to construct a 6D confidence ellipsoid.
• Defining a parametric space with six dual Rodrigues parameters to obtain a 6x6 covariance matrix and the ellipsoid.
• Applying an inverse operation to recover rotation matrices and translation vectors from the ellipsoid points.

Main Results:

• The dual-Rodrigues formulation enables the computation of a 6D confidence ellipsoid.
• Comparison with the RTCL method shows the proposed approach yields more compact and effective swept volumes.
• The method demonstrates particular efficacy in handling screw displacements.

Conclusions:

• The dual-Rodrigues formulation provides a superior method for calculating kinematic confidence regions.
• This approach offers advantages over traditional methods, especially for complex spatial displacements.
• The findings contribute to more precise uncertainty quantification in robotics and spatial analysis.