Invariant errors of discrete motion constrained by actual kinematic pairs

- Present a pioneering approach to handle error-induced motion of an actual joint.
- Provide the theoretical basis of discrete error motion by differential geometry.
- Develop a novel scheme with invariants and their errors by saddle point program.
- Propose the global invariant errors to evaluate inaccuracies of an actual R-joint.
- Carry out experiments on a spindle to validate the proposed scheme.

Motion of a rigid body is systematically investigated using invariants of line-trajectories, and the invariant errors are proposed for the first time to evaluate accuracy of motion for three actual joints C, H and R. A general spatial motion of a rigid body can be dissolved into the following motion with a reference line, having four DOFs, and the relative motion about and along the reference line, having two additional DOFs. The necessary and sufficient conditions of cylindrical motion, helical motion and rotational motion in both continuous and discrete error forms are respectively derived by invariants of line-trajectories and global invariants with minimal values in differential geometry. For discrete data sets, a novel scheme based on the invariants and their fitting errors obtained by the saddle point programming, is developed to characterize the nominal motion and the error-induced motion. The invariant errors are presented to quantify the accuracy of the discrete error motion of joints C, H and R. Experiment was carried out on a machine tool spindle to demonstrate the advantages of the proposed invariants-based error evaluation scheme. The scheme provides a new method for quantifying accuracy of motion and improving performances of machine tools and robots.