Correcting Duporcq's theorem

Area of Science:

• Robotics
• Kinematics
• Mechanical Engineering

Background:

• Ernest Duporcq's 1898 theorem on rigid-body motions with spherical trajectories lacked rigorous proof.
• The theorem is relevant to understanding self-motions in planar Stewart-Gough platforms.

Purpose of the Study:

• To analyze Duporcq's theorem in the context of planar Stewart-Gough platform self-motions.
• To demonstrate the theorem's inaccuracies.
• To present a corrected version of the theorem.

Main Methods:

• Theoretical analysis of rigid-body kinematics.
• Investigation of spherical trajectories.
• Examination of Stewart-Gough platform configurations.

Main Results:

• Duporcq's theorem, as originally stated, is not correct.
• Specific conditions under which the theorem fails are identified.
• A revised and accurate theorem is formulated.

Conclusions:

• The original Duporcq theorem requires correction for accurate application to planar Stewart-Gough platforms.
• The revised theorem provides a more precise understanding of rigid-body motions and platform self-motions.