A Novel Contouring Control Method Based on Optimal Vector-Referenced Moving Frame for 3D Trajectory With Zero Curvature

Contouring control of 3D trajectory is critical in multi-axial machine tool, scanning stage and other precision automation systems. Currently, most contouring controllers are based on Frenet frames, thus their limited applicability to nonzero-curvature 3D trajectories rather than zero-curvature ones which exist ubiquitously in multi-axial motion systems. This paper proposes an optimal vector-referenced moving frame based contouring controller (OVRMFCC) suitable for contouring control of 3D zero-curvature trajectories. Firstly, the optimal vector-referenced moving frame (OVRMF) capable of framing arbitrary finite-length smooth 3D trajectory regardless of its curvature was proposed. Then, a contouring controller (OVRMFCC) based on OVRMF was designed, followed by derivation of its analytical form and proof of its convergence. Finally, this controller was deployed to an FPGA-based controller target with comprehensive comparison experiments on a triaxial system. Experimental results indicates that OVRMFCC reduces at least 46.6% maximum contour error, and 25.0% root-mean-square contour error compared to cross-coupled controller. Besides, OVRMFCC achieves almost the same precision on trajectories with nonzero curvature or curvature singularities compared to TCF. It still maintains high-precision contour tracking on trajectories with continuous zero-curvature segments or planned discrete trajectories with sharp curvature changes, while TCF crashes or leads to several times larger contour error. Note to Practitioners—This work is motivated by the increasing need of contouring control of 3D trajectory in precision automation systems. The mainstream 3D contouring controllers, like task coordinated frame method and model predicted contouring controller, are invalid for zero-curvature 3D trajectory which ubiquitously exists in motion system since they are based on Frenet frame which fails to be defined where curvature is zero. Although cross-coupled controller can tackle zero-curvature 3D trajectory, it proves inefficient in reducing contour error as it is commonly model-free and not specifically designed. To solve this problem, we proposed an optimal vector referenced moving frame (OVRMF) for arbitrary infinite-length smooth 3D trajectory framing and proves its existence strictly. The main advantage of OVRMF lies in its independence of curvature, thus its existence everywhere as long as the curve is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C^{1}$ </tex-math></inline-formula> continuous. And based on OVRMF a contouring controller (OVRMFCC) is designed which combines the benefit of both TCF and OVRMF. With this, OVRMFCC can track any infinite-length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C^{3}$ </tex-math></inline-formula>-continuous 3D trajectory, which fill the gap of traditional 3D contouring controller. Experimental results reveal that OVRMFCC maintains high precision regardless of trajectory curvature, which outperforms both cross-coupled controller and TCF.